DIMACS

Technical

Report

July

Packing

of

induced

stars

in

a

graph

by

Alexander

K.

RUTCOR

Rutgers

University

New

Brunswick

New

Jersey

and

Department

of

Mathematics

University

of

Puerto

Rico

P.O.

Box

San

Juan

PR

-LCB-

DIMACS

is

a

partnership

of

Rutgers

University

Princeton

University

AT&T

Research

Bellcore

and

Bell

Laboratories

DIMACS

is

an

NSF

Science

and

Technology

Center

funded

under

contract

STC

-LCB-

-LCB-

and

also

receives

support

from

the

New

Jersey

Commission

on

Science

and

Technology

Abstract

We

consider

simple

undirected

graphs

An

edge

subset

A

of

G

is

called

an

induced

n

-LCB-

star

packing

of

G

if

every

component

of

the

subgraph

G

-LSB-

A

-RSB-

induced

by

A

is

a

star

with

at

most

n

edges

and

is

an

induced

subgraph

of

G

We

consider

the

problem

of

finding

an

induced

n

-LCB-

star

packing

of

G

that

covers

the

maximum

number

of

vertices

This

problem

is

a

natural

generalization

of

the

classical

matching

problem

We

show

that

many

classical

results

on

matchings

-LRB-

such

as

the

Tutte

Theorem

the

Berge

Duality

Theorem

the

Gallai

-LCB-

Edmonds

Structure

Theorem

the

Matching

Matroid

Theorem

-RRB-

can

be

extended

to

induced

n

-LCB-

star

packings

in

a

graph

author

gratefully

acknowledges

DIMACS

for

the

support

during

the

visit

when

this

paper

was

prepared

Introduction

The

problem

of

finding

a

maximum

matching

in

a

graph

is

well

known

Many

interesting

results

-LRB-

that

are

classical

now

-RRB-

concerning

this

problem

have

been

found

by

Tutte

Edmonds

Berge

Gallai

Fulkerson

Lovasz

and

many

others

-LSB-

-RSB-

One

of

the

ideas

to

generalize

the

matching

theory

is

to

try

to

replace

the

edges

in

a

matching

by

some

subgraphs

of

prescribed

type

Let

G

be

a

graph

and

F

be

a

family

of

subgraphs

of

G

An

edge

subset

A

of

G

is

called

an

F

-LCB-

packing

of

G

if

every

component

of

the

subgraph

G

-LSB-

A

-RSB-

induced

by

A

in

G

is

a

graph

in

the

family

F

The

F

-LCB-

packing

problem

consists

of

finding

an

F

-LCB-

packing

A

of

G

that

covers

the

maximum

number

of

vertices

If

F

consists

of

all

-LCB-

cliques

-LRB-

all

subgraphs

of

one

edge

-RRB-

of

G

then

the

F

-LCB-

packing

problem

is

precisely

the

classical

maximum

matching

problem

The

F

-LCB-

packing

problem

has

extensively

been

studied

by

many

authors

for

different

classes

of

families

F

Comprehansive

surveys

of

results

obtained

so

far

can

be

found

in

-LSB-

-RSB-

Clearly

the

properties

of

the

F

-LCB-

packing

problem

essentially

depend

on

the

family

F

It

is

not

surprising

that

the

problem

turns

out

to

be

NP

-LCB-

complete

for

most

of

the

families

F

-LRB-

e.g.

-LSB-

-RSB-

-RRB-

Surprisingly

the

problem

can

be

solved

in

polynomial

time

for

some

non

-LCB-

trivial

classes

of

families

F

and

many

important

results

in

matching

theory

can

be

generalized

in

those

cases

-LRB-

e.g.

-LSB-

-RSB-

-RRB-

Let

F

be

the

set

of

stars

of

G

with

at

least

one

and

at

most

n

edges

Then

we

have

a

star

packing

problem

Sn

This

problem

is

one

of

the

simplest

generalizations

of

the

matching

problem

that

turns

out

to

be

good

-LSB-

-RSB-

-LRB-

this

also

follows

easily

from

the

approach

developed

in

this

paper

-RRB-

The

nature

of

most

of

the

known

good

F

-LRB-

G

-RRB-

-LCB-

packing

problems

are

similar

to

that

for

the

star

packing

problem

In

this

paper

we

consider

a

star

packing

problem

with

an

additional

condition

A

new

requirment

is

that

every

star

in

an

F

-LCB-

packing

should

be

an

induced

subgraph

of

G

In

other

words

we

consider

an

F

-LCB-

packing

problem

ISn

where

F

Fn

-LRB-

G

-RRB-

is

the

set

of

all

induced

stars

of

G

having

at

most

n

edges

Main

results

are

described

in

Section

In

Section

we

introduce

the

notions

of

alternating

and

augmenting

trails

as

well

as

passive

paths

and

active

trails

and

describe

some

important

properties

of

such

trails

Analysis

of

induced

star

packings

without

augmenting

trails

is

given

in

Section

Duality

Theorem

-LRB-

Section

-RRB-

the

matroid

results

-LRB-

Section

-RRB-

and

Structure

Theorem

-LRB-

Section

-RRB-

on

induced

star

packings

are

easy

consequences

of

this

analysis

In

Section

we

give

a

polynomial

time

algorithm

for

solving

the

induced

star

packing

problem

in

a

graph

The

results

of

the

paper

were

presented

at

RUTCOR

Seminar

Rutgers

University

in

at

the

Southeastern

International

Conference

on

Combinatorics

Graph

Theory

and

Computing

in

February

at

the

Seymour

-LCB-

Schrijver

Workshop

on

disjoint

paths

in

DIMACS

in

November

and

at

seminars

in

MIT

University

of

Waterloo

Simon

Fraiser

University

MacMaster

University

University

of

Quebec

in

Montreal

and

other

institutions

Main

results

The

notion

and

facts

on

graphs

and

matroids

that

are

used

but

not

described

here

can

be

found

in

-LSB-

-RSB-

and

-LSB-

-RSB-

respectively

We

consider

undiracted

graphs

without

loops

or

parallel

edges

Let

V

-LRB-

G

-RRB-

and

E

-LRB-

G

-RRB-

denote

the

sets

of

vertices

and

edges

of

a

graph

G

respectively

Given

an

edge

subset

A

of

G

let

G

-LSB-

A

-RSB-

denote

the

subgraph

of

G

induced

by

A

i.e.

E

-LRB-

G

-LSB-

A

-RSB-

-RRB-

A

and

V

-LRB-

G

-LSB-

A

-RSB-

-RRB-

consists

of

the

vertices

of

G

covered

by

A

-LRB-

or

incident

to

at

least

one

edge

in

A

-RRB-

Given

a

vertex

subset

B

of

G

let

G

-LSB-

B

-RSB-

-LRB-

or

sometimes

simply

B

-RRB-

denote

the

subgraph

of

G

induced

by

B

i.e.

G

-LSB-

B

-RSB-

Gn

-LRB-

V

-LRB-

G

-RRB-

nB

-RRB-

An

induced

subgraph

of

G

is

a

subgraph

induced

by

a

vertex

subset

of

G

In

other

words

a

subgraph

of

G

is

induced

if

it

can

be

obtained

from

G

by

deleting

some

vertices

of

G

We

write

Z

-LSB-

X

-RSB-

instead

of

Z

-LRB-

G

-LSB-

X

-RSB-

-RRB-

for

example

V

-LSB-

X

-RSB-

instead

of

V

-LRB-

G

-LSB-

X

-RSB-

-RRB-

E

-LSB-

X

-RSB-

instead

of

E

-LRB-

G

-LSB-

X

-RSB-

-RRB-

etc.

Given

a

subgraph

H

and

an

edge

subset

U

of

G

we

put

U

-LRB-

H

-RRB-

U

E

-LRB-

H

-RRB-

A

star

S

is

a

connected

graph

in

which

all

the

edges

have

a

common

vertex

When

considering

the

problem

ISn

of

packing

induced

stars

in

a

graph

we

need

the

following

notion

We

say

that

an

edge

set

A

of

G

is

an

induced

n

-LCB-

star

packing

of

G

-LRB-

or

simply

an

induced

n

-LCB-

packing

of

G

-RRB-

if

every

component

S

of

G

-LSB-

A

-RSB-

is

an

induced

subgraph

of

G

isomorphic

to

a

star

with

at

most

n

edges

An

induced

n

-LCB-

star

packing

A

is

called

perfect

if

V

-LSB-

A

-RSB-

V

-LRB-

G

-RRB-

i.e.

if

A

covers

all

the

vertices

of

the

graph

Clearly

if

n

is

at

least

the

maximum

vertex

degree

of

G

then

there

are

actually

no

restriction

on

the

size

of

stars

in

an

induced

n

-LCB-

star

packing

and

so

in

this

case

an

induced

n

-LCB-

star

packing

of

G

will

be

called

an

induced

star

packing

When

considering

the

problem

Sn

of

packing

stars

-LRB-

not

necessarily

induced

-RRB-

in

a

graph

we

will

have

the

corresponding

notions

of

an

n

-LCB-

star

packing

-LRB-

or

simply

n

-LCB-

packing

-RRB-

a

perfect

n

-LCB-

star

packing

star

packing

and

perfect

star

packing

Let

PISn

-LRB-

G

-RRB-

denote

the

set

of

induced

n

-LCB-

star

packings

of

G

The

problem

ISn

we

are

going

to

consider

is

to

find

an

induced

n

-LCB-

packing

A

in

G

that

covers

the

maximum

number

of

vertices

i.e.

jV

-LSB-

A

-RSB-

j

maxfV

-LSB-

X

-RSB-

X

PISn

-LRB-

G

-RRB-

g

Such

an

induced

n

-LCB-

packing

A

in

G

is

called

vertex

maximum

or

simply

V

-LCB-

maximum

In

particular

we

want

to

characterize

graphs

having

a

perfect

induced

n

-LCB-

packing

Clearly

-LRB-

as

well

as

-RRB-

is

a

matching

problem

Unless

otherwise

is

stated

explicitely

we

assume

throughout

the

paper

that

n

is

an

integer

at

least

A

graph

G

is

called

ISn

-LCB-

critical

if

G

has

no

perfect

induced

n

-LCB-

packing

but

G

n

x

has

a

perfect

induced

n

-LCB-

packing

for

every

vertex

x

of

G

An

graph

is

called

usually

matching

-LCB-

critical

or

hypomatchable

Let

crn

-LRB-

G

-RRB-

denote

the

number

of

ISn

-LCB-

critical

components

of

G.

Let

W

-LSB-

A

-RSB-

denote

the

set

of

vertices

of

G

that

are

not

covered

by

A

i.e.

W

-LSB-

A

-RSB-

V

-LRB-

G

-RRB-

n

V

-LSB-

A

-RSB-

It

turns

out

that

the

following

duality

theorem

holds

Theorem

Let

n

be

a

positive

integer

Then

minfjW

-LSB-

A

-RSB-

j

A

PISn

-LRB-

G

-RRB-

g

maxfcrn

-LRB-

G

n

X

-RRB-

njXj

X

V

-LRB-

G

-RRB-

g.

For

n

this

is

a

well

known

result

-LSB-

-RSB-

It

is

easy

to

see

that

the

problem

of

finding

an

induced

n

-LCB-

star

packing

which

covers

at

least

a

given

number

of

vertices

in

a

graph

belongs

to

the

class

NP

A

natural

question

is

whether

this

problem

also

belongs

to

CoNP

In

other

words

we

would

like

to

know

whether

there

exists

-LRB-

and

we

can

find

-RRB-

a

polynomial

time

certificate

for

a

graph

not

to

have

an

induced

n-star

packing

that

covers

a

given

number

of

vertices

The

above

theorem

would

provide

such

certificate

if

ISn

-LCB-

critical

graphs

could

be

recognized

in

polynomial

time

Therefore

it

is

useful

to

have

a

good

characterization

of

ISn

-LCB-

critical

graphs

A

graph

is

called

IS-critical

if

it

is

ISn

-LCB-

critical

for

every

integer

n

A

graph

F

is

called

an

odd

clique

tree

if

F

is

connected

and

every

block

of

F

is

a

complete

graph

with

an

odd

number

of

vertices

We

prove

that

Theorem

The

following

conditions

are

equivalent

-LRB-

-RRB-

F

is

an

ISn

-LCB-

critical

graph

n

-LRB-

-RRB-

F

is

a

IS

-LCB-

critical

graph

and

-LRB-

-RRB-

F

is

an

odd

clique

tree

Let

oct

-LRB-

G

-RRB-

denote

the

number

of

components

of

G

which

are

odd

clique

trees

From

Theorem

we

have

crn

-LRB-

G

-RRB-

oct

-LRB-

G

-RRB-

for

every

integer

n

and

so

crn

-LRB-

G

-RRB-

does

not

depend

on

n

Therefore

from

Theorems

and

we

obtain

a

Duality

Theorem

that

provides

a

good

characterization

of

the

problem

Theorem

Let

n

Then

minfjW

-LSB-

A

-RSB-

j

A

PISn

-LRB-

G

-RRB-

g

maxfoct

-LRB-

G

n

X

-RRB-

njXj

X

V

-LRB-

G

-RRB-

g

This

theorem

is

analogous

to

the

Berge

matching

theorem

-LSB-

-RSB-

From

Theorems

and

it

follows

that

the

problem

of

finding

an

induced

n-star

packing

that

covers

at

least

a

given

number

of

vertices

in

a

graph

belongs

to

CoNP

The

above

Duality

Theorem

provides

in

particular

a

criterion

for

a

graph

to

have

a

perfect

induced

n

-LCB-

packing

which

is

analogous

to

the

Tutte

matching

theorem

-LSB-

-RSB-

Theorem

A

graph

G

has

a

perfect

induced

n

-LCB-

packing

if

and

only

if

oct

-LRB-

G

n

X

-RRB-

njXj

for

every

X

V

-LRB-

G

-RRB-

From

Theorem

we

have

Theorem

A

connected

graph

G

does

not

have

a

perfect

induced

star

packing

if

and

only

if

G

is

an

odd

clique

tree

The

next

theorem

is

analogous

to

the

Gallai

-LCB-

Edmonds

Structure

theorem

on

matchings

in

a

graph

-LSB-

-RSB-

We

use

the

following

notation

Cn

-LRB-

G

-RRB-

is

the

set

of

all

vertices

of

G

which

are

not

covered

by

at

least

one

V

-LCB-

maximum

induced

n

-LCB-

packing

of

G

Hn

-LRB-

G

-RRB-

is

the

set

of

all

vertices

in

V

-LRB-

G

-RRB-

n

Cn

-LRB-

G

-RRB-

adjacent

to

at

least

one

vertex

in

Cn

-LRB-

G

-RRB-

and

Dn

-LRB-

G

-RRB-

V

-LRB-

G

-RRB-

n

-LRB-

Hn

-LRB-

G

-RRB-

-LSB-

Cn

-LRB-

G

-RRB-

-RRB-

Let

Dn

-LRB-

G

-RRB-

and

Cn

-LRB-

G

-RRB-

denote

subgraphs

of

G

induced

by

the

vertex

sets

Dn

-LRB-

G

-RRB-

and

Cn

-LRB-

G

-RRB-

respectively

A

matching

of

G

is

called

nearly

perfect

if

it

covers

all

but

exactly

one

vertex

of

G.

C

-RRB-

-LRB-

G

D

-LRB-

G

-RRB-

H

-LRB-

G

-RRB-

n

n

n

Figure

The

structure

of

a

graph

with

respect

to

its

induced

n

-LCB-

packings

Theorem

Let

G

be

a

graph

and

n

be

an

integer

at

least

Then

-LRB-

-RRB-

the

components

of

the

subgraph

Cn

-LRB-

G

-RRB-

are

ISn

-LCB-

critical

-LRB-

are

odd

clique

trees

-RRB-

-LRB-

-RRB-

the

subgraph

Dn

-LRB-

G

-RRB-

has

a

perfect

induced

n

-LCB-

packing

-LRB-

-RRB-

if

A

is

a

maximum

induced

n

-LCB-

packing

of

G

then

it

contains

-LRB-

-RRB-

a

near

perfect

matching

of

each

component

of

the

subgraph

Cn

-LRB-

G

-RRB-

-LRB-

-RRB-

a

perfect

induced

n

-LCB-

packing

of

the

subgraph

Dn

-LRB-

G

-RRB-

and

-LRB-

-RRB-

a

set

of

jHn

-LRB-

G

-RRB-

j

disjoint

n

-LCB-

stars

such

that

their

heads

are

in

Hn

-LRB-

G

-RRB-

their

tails

are

in

Cn

-LRB-

G

-RRB-

and

each

component

of

Cn

-LRB-

G

-RRB-

contains

at

most

one

tail

of

all

these

stars

-LRB-

-RRB-

W

-LSB-

A

-RSB-

minfjW

-LSB-

T

-RSB-

j

T

PISn

-LRB-

G

-RRB-

g

oct

-LRB-

G

n

Hn

-LRB-

G

-RRB-

-RRB-

njHn

-LRB-

G

-RRB-

j

This

theorem

is

illustrated

in

Figure

where

all

rectangles

in

Cn

-LRB-

G

-RRB-

are

all

components

-LRB-

odd

clique

trees

-RRB-

of

Cn

-LRB-

G

-RRB-

the

thin

lines

are

edges

in

E

-LRB-

G

-RRB-

n

A

and

the

thick

lines

are

edges

in

A.

Let

Z

be

a

vertex

subset

of

G

such

that

minfjW

-LSB-

A

-RSB-

j

A

PISn

-LRB-

G

-RRB-

g

oct

-LRB-

G

n

Z

-RRB-

njZj

We

call

such

subset

Z

an

ISn

-LCB-

obstacle

in

G

because

Z

provides

a

tight

upper

bound

on

the

number

of

vertices

in

G

that

can

be

covered

by

an

induced

n

-LCB-

packing

We

prove

the

following

Theorem

Let

X

and

Y

be

IS

-LCB-

obstacles

in

G

Then

X

-LSB-

Y

and

X

Y

are

also

IS

-LCB-

obstacles

in

G

This

theorem

is

similar

to

that

for

the

matchings

in

a

graph

Let

Zn

-LRB-

G

-RRB-

denote

the

set

of

all

ISn

-LCB-

obstacles

in

G.

By

using

Theorems

and

we

prove

Theorem

Let

Ln

-LRB-

G

-RRB-

-LRB-

Zn

-LRB-

G

-RRB-

-RRB-

denote

the

set

Zn

-LRB-

G

-RRB-

partially

ordered

by

the

inclusion

operation

Then

-LRB-

-RRB-

Ln

-LRB-

G

-RRB-

is

a

sublattice

of

of

the

lattice

of

all

subsets

of

V

-LRB-

G

-RRB-

under

inclusion

-LRB-

-RRB-

Hn

-LRB-

G

-RRB-

is

the

minimum

element

of

Ln

-LRB-

G

-RRB-

and

-LRB-

-RRB-

if

X

Ln

-LRB-

G

-RRB-

then

X

n

Hn

-LRB-

G

-RRB-

Ln

-LRB-

Dn

-LRB-

G

-RRB-

-RRB-

An

augmenting

A

-LCB-

trail

is

a

trail

P

of

G

such

that

the

edges

of

A

and

E

-LRB-

G

-RRB-

n

A

alternate

in

P

-LRB-

P

-RRB-

is

an

induced

n

-LCB-

packing

and

jV

-LSB-

A

-RSB-

j

V

-LSB-

-LRB-

P

-RRB-

-RSB-

We

show

that

augmenting

A

-LCB-

trails

are

sufficient

tools

to

find

a

maximum

induced

n

-LCB-

packing

Theorem

An

induced

n

-LCB-

packing

A

of

G

covers

the

maximum

number

of

vertices

of

G

if

and

only

if

there

is

no

augmenting

A

-LCB-

trail

in

G

This

theorem

is

similar

to

that

for

the

matchings

in

a

graph

-LSB-

-RSB-

Let

ISn

-LRB-

G

-RRB-

denote

the

set

of

vertex

subsets

X

of

G

such

that

X

can

be

covered

by

an

induced

n

-LCB-

packing

of

G

It

is

easy

to

see

that

ISn

-LRB-

G

-RRB-

has

the

hereditary

property

Analysis

of

the

properties

of

induced

n

-LCB-

packings

in

a

graph

without

augmenting

A

-LCB-

trails

enables

us

to

give

a

description

of

a

cycle

of

the

hereditary

family

ISn

-LRB-

G

-RRB-

of

vertex

sets

of

G

-LRB-

see

below

-RRB-

By

using

this

description

we

prove

the

following

theorem

Put

MISn

-LRB-

G

-RRB-

-LRB-

V

-LRB-

G

-RRB-

ISn

-LRB-

G

-RRB-

-RRB-

An

element

of

a

matroid

M

is

called

cyclic

if

it

belongs

to

at

least

one

circuit

of

M

and

acyclic

or

a

coloop

otherwise

Theorem

Let

n

be

an

integer

n

Then

-LRB-

-RRB-

MISn

-LRB-

G

-RRB-

is

a

matroid

with

the

independence

set

ISn

-LRB-

G

-RRB-

-LRB-

-RRB-

Cn

-LRB-

G

-RRB-

is

the

set

of

cyclic

elements

of

the

matroid

MISn

-LRB-

G

-RRB-

and

-LRB-

-RRB-

Hn

-LRB-

G

-RRB-

-LSB-

Dn

-LRB-

G

-RRB-

is

the

set

of

coloops

of

MISn

-LRB-

G

-RRB-

Note

that

-LRB-

G

-RRB-

is

the

well-known

matching

matroid

of

G

-LSB-

-RSB-

Theorem

can

also

be

proved

by

using

Structure

Theorem

An

edge

subset

A

of

G

is

called

an

k

-LCB-

induced

n

-LCB-

packings

if

every

component

of

G

-LSB-

A

-RSB-

having

at

least

k

edges

is

an

induced

subgraph

of

G.

Let

PISkn

-LRB-

G

-RRB-

denote

the

set

of

all

k

-LCB-

induced

n

-LCB-

packings

of

G

and

let

ISkn

-LRB-

G

-RRB-

denote

the

sets

of

vertex

subsets

X

of

G

such

that

X

is

covered

by

a

k

-LCB-

induced

n

-LCB-

packing

k

In

particular

-LRB-

G

-RRB-

PISn

-LRB-

G

-RRB-

is

the

set

of

all

induced

n

-LCB-

star

packings

of

G

and

PISkn

-LRB-

G

-RRB-

PSn

-LRB-

G

-RRB-

is

the

set

of

all

n

-LCB-

star

packings

of

G

for

k

n.

Theorem

ISn

-LRB-

G

-RRB-

ISkn

-LRB-

G

-RRB-

for

every

integer

k

and

n

such

that

k

n

This

theorem

follows

directly

from

the

following

simple

Claim

Let

F

be

a

graph

having

a

vertex

adjacent

to

every

other

vertex

in

F

Then

F

has

a

matching

M

such

that

if

F

is

an

odd

clique

then

F

n

F

-LSB-

M

-RSB-

is

a

triangle

otherwise

F

n

F

-LSB-

M

-RSB-

is

an

induced

star

in

F

The

approach

developed

below

for

obtaining

the

above

results

also

gives

natural

proofs

of

the

following

results

Let

PSn

-LRB-

G

-RRB-

denote

the

set

of

n

-LCB-

packings

of

G

and

isv

-LRB-

G

-RRB-

denote

the

number

of

isolated

vertices

of

G.

Theorem

Let

n

be

an

integer

n

Then

minfjW

-LSB-

A

-RSB-

j

A

PSn

-LRB-

G

-RRB-

g

maxfisv

-LRB-

G

n

X

-RRB-

njXj

X

V

-LRB-

G

-RRB-

g.

From

the

above

theorem

we

have

in

particular

Theorem

-LSB-

-RSB-

A

graph

G

has

a

perfect

n

-LCB-

packing

if

and

only

if

isv

-LRB-

G

n

X

-RRB-

njXj

for

every

X

V

-LRB-

G

-RRB-

Let

Sn

-LRB-

G

-RRB-

be

the

set

of

vertex

subsets

X

of

G

such

that

X

can

be

covered

by

an

n

-LCB-

packing

of

G.

Put

MSn

-LRB-

G

-RRB-

-LRB-

V

-LRB-

G

-RRB-

Sn

-LRB-

G

-RRB-

-RRB-

Theorem

-LSB-

-RSB-

MSn

-LRB-

G

-RRB-

is

a

matroid

with

the

independence

set

Sn

-LRB-

G

-RRB-

for

every

integer

n

This

theorem

also

follows

immediately

from

the

simple

fact

that

MSn

-LRB-

G

-RRB-

is

the

union

of

n

matching

matroids

-LRB-

G

-RRB-

The

same

approach

can

be

used

to

prove

the

following

theorem

on

n

-LCB-

star

packings

analogous

to

the

Gallai

-LCB-

Edmonds

Structure

theorem

on

matchings

in

a

graph

-LSB-

-RSB-

and

to

Theorem

on

induced

n

-LCB-

star

packings

in

a

graph

We

use

the

following

notation

C

n

-LRB-

G

-RRB-

is

the

set

of

all

vertices

of

G

which

are

not

covered

by

at

least

one

maximum

n

-LCB-

packing

of

G

H

n

-LRB-

G

-RRB-

is

the

set

of

all

vertices

in

V

-LRB-

G

-RRB-

nC

n

-LRB-

G

-RRB-

adjacent

to

at

least

one

vertex

in

C

n

-LRB-

G

-RRB-

-LRB-

G

-RRB-

V

-LRB-

G

-RRB-

n

-LRB-

H

n

-LRB-

G

-RRB-

-LSB-

C

n

-LRB-

G

-RRB-

Let

-LRB-

G

-RRB-

and

C

n

-LRB-

G

-RRB-

denote

subgraphs

of

G

induced

by

the

vertex

sets

-LRB-

G

-RRB-

and

C

n

-LRB-

G

-RRB-

respectively

Theorem

Let

G

be

a

graph

and

n

be

an

integer

at

least

Then

-LRB-

-RRB-

the

components

of

the

subgraph

C

n

-LRB-

G

-RRB-

are

Sn

-LCB-

critical

-LRB-

are

isolated

vertices

-RRB-

-LRB-

-RRB-

the

subgraph

-LRB-

G

-RRB-

has

a

perfect

n

-LCB-

packing

-LRB-

-RRB-

if

A

is

a

maximum

n

-LCB-

packing

of

G

then

it

contains

-LRB-

-RRB-

a

perfect

n

-LCB-

packing

of

the

subgraph

-LRB-

G

-RRB-

and

-LRB-

-RRB-

a

set

of

jH

n

-LRB-

G

-RRB-

j

disjoint

n

-LCB-

stars

such

that

their

heads

are

in

H

n

-LRB-

G

-RRB-

their

tails

are

in

C

n

-LRB-

G

-RRB-

-LRB-

-RRB-

W

-LSB-

A

-RSB-

minfjV

-LRB-

G

-RRB-

n

V

-LSB-

Y

-RSB-

j

Y

PSn

-LRB-

G

-RRB-

g

oct

-LRB-

G

n

H

n

-LRB-

G

-RRB-

-RRB-

njH

n

-LRB-

G

-RRB-

j.

Theorem

Let

G

be

a

graph

and

n

be

an

integer

Then

C

n

-LRB-

G

-RRB-

and

H

n

-LRB-

G

-RRB-

-LSB-

-LRB-

G

-RRB-

are

the

set

of

cyclic

elements

and

the

set

of

coloops

of

the

matroid

MSn

-LRB-

G

-RRB-

respectively

By

using

arguments

similar

to

that

in

our

proofs

we

can

find

polynomial

time

algorithms

for

solving

the

corresponding

problems

-LRB-

see

Section

-RRB-

So

we

have

Theorem

The

following

problems

can

be

solved

in

polynomial

time

-LRB-

P

-RRB-

find

an

induced

n

-LCB-

packing

in

a

graph

covering

the

maximum

number

of

vertices

-LRB-

P

-RRB-

recognize

ISn

-LCB-

critical

graphs

-LRB-

P

-RRB-

recognize

a

graph

having

a

perfect

induced

n

-LCB-

packing

and

-LRB-

P

-RRB-

recognize

a

vertex

subset

of

G

that

can

be

covered

by

an

induced

n-star

packing

It

is

known

that

some

combinatorial

optimization

problems

for

matroids

can

be

solved

in

a

polynomial

number

of

calls

to

the

independence

oracle

of

the

matroid

By

Theorem

-LRB-

-RRB-

the

independence

oracle

for

the

matroid

MISn

-LRB-

G

-RRB-

can

be

realized

by

a

polynomial

time

algorithm

Therefore

the

corresponding

combinatorial

optimization

problems

for

MISn

-LRB-

G

-RRB-

can

be

solved

in

polynomial

time

In

particular

we

have

Theorem

The

following

problems

can

be

solved

in

polynomial

time

-LRB-

P

-RRB-

find

the

maximum

number

of

disjoint

maximum

induced

n

-LCB-

packings

in

a

graph

-LRB-

P

-RRB-

find

the

minimum

number

of

induced

n

-LCB-

packings

that

covers

all

the

vertices

of

a

graph

-LRB-

P

-RRB-

given

a

matroid

M

on

V

-LRB-

G

-RRB-

find

an

induced

n

-LCB-

packing

of

G

that

covers

an

independent

set

of

M

of

maximum

size

and

-LRB-

P

-RRB-

given

non-negative

weights

of

the

vertices

of

G

find

an

induced

n

-LCB-

packing

of

G

that

covers

a

vertex

set

of

G

of

maximum

total

weight

Alternating

and

augmenting

trails

The

concepts

of

alternating

and

augmenting

paths

play

a

very

important

role

in

the

matching

theory

-LSB-

-RSB-

In

we

have

introduced

analogous

concepts

for

a

matroid

which

is

the

union

of

several

matroids

We

have

considered

special

types

of

these

paths

-LRB-

so

called

passive

and

active

paths

-RRB-

We

used

the

idea

of

a

passive

path

reachability

of

a

covered

element

from

a

non-covered

element

to

give

a

simple

algorithmic

proof

of

the

main

theorem

in

matroid

optimization

on

the

rank

of

the

union

of

matroids

and

to

give

a

polynomial

time

algorithms

for

solving

packing

covering

and

intersection

problems

in

matroids

-LRB-

described

in

-LSB-

-RSB-

-RRB-

In

-LSB-

-RSB-

this

approach

has

been

used

to

find

a

fastest

known

algorithm

for

solving

these

matroid

optimization

problems

In

this

section

we

are

going

to

introduce

similar

concepts

of

alternating

and

augmenting

paths

and

in

particular

passive

and

active

paths

and

to

use

similar

approach

for

analysing

the

induced

n

-LCB-

star

packing

problem

By

using

passive

alternating

paths

and

the

corresponding

reachability

we

will

investigate

the

properties

of

a

vertex

maximal

induced

n

-LCB-

packing

The

proofs

of

the

main

results

described

in

the

previous

section

are

natural

byproducts

of

this

analysis

A

trail

P

in

a

graph

G

is

a

sequence

xk

where

each

xj

is

a

vertex

and

each

ei

is

an

edge

of

G

each

ei

-LRB-

xi

xi

-RRB-

and

ei

ej

for

i

j

If

in

addition

xi

xj

for

i

j

then

such

trails

is

called

a

path

in

G

Since

G

has

no

loops

or

parallel

edges

we

can

also

describe

a

trail

by

the

sequence

of

its

vertices

P

xk

Note

that

xk

xk

and

xk

xk

are

different

trails

A

subtrail

xiPxj

i

j

of

the

trail

is

a

subsequence

xiejxi

ej

of

the

sequence

xk

A

subpath

of

a

path

is

defined

similarly

Let

A

be

an

induced

n

-LCB-

packing

of

G

A

trail

P

xk

k

is

called

an

A

-LCB-

alternating

trail

or

simply

an

A

-LCB-

trail

of

G

if

jfei

ei

g

Aj

for

every

i

k

We

recall

that

if

X

and

Y

are

sets

then

-LRB-

X

n

Y

-RRB-

-LSB-

-LRB-

Y

n

X

-RRB-

We

are

interested

in

A

-LCB-

alternating

trails

P

such

that

if

A

is

an

induced

n

-LCB-

packing

then

-LRB-

P

-RRB-

is

also

an

induced

n

-LCB-

packing

in

G.

A

-LCB-

trail

P

is

said

to

be

augmenting

if

-LRB-

P

-RRB-

is

an

induced

n

-LCB-

star

packing

and

jV

-LSB-

A

-RSB-

j

jV

-LSB-

-RSB-

j

-LRB-

clearly

in

this

case

V

-LSB-

A

-RSB-

ae

V

-LSB-

-RSB-

-RRB-

Let

S

-LRB-

A

-RRB-

denote

the

set

of

stars

of

A

i.e.

the

set

of

components

of

G

-LSB-

A

-RSB-

A

star

S

of

A

is

big

if

jE

-LRB-

S

-RRB-

j

n

small

if

jE

-LRB-

S

-RRB-

j

and

intermediate

if

jE

-LRB-

S

-RRB-

j

n

A

k

-LCB-

star

is

a

star

with

k

edges

Let

Ak

denote

the

set

of

edges

in

A

belonging

to

the

k

-LCB-

stars

of

A

i.e.

Ak

-LSB-

fE

-LRB-

S

-RRB-

S

Sk

-LRB-

A

-RRB-

g.

Small

and

big

stars

and

respectively

and

An

will

play

an

essential

role

in

our

consideration

If

S

is

a

non-small

star

then

the

common

vertex

of

all

edges

of

S

is

called

the

head

of

S

and

a

vertex

of

S

distinct

from

the

head

of

S

-LRB-

a

vertex

of

degree

in

S

-RRB-

is

called

a

tail

of

S.

Let

h

-LRB-

S

-RRB-

and

T

-LRB-

S

-RRB-

denote

the

head

and

the

set

of

tails

of

S

respectively

An

A

-LCB-

trail

xk

k

is

called

a

passive

A

-LCB-

path

if

the

following

conditions

hold

-LRB-

-RRB-

P

is

an

A

-LCB-

path

-LRB-

-RRB-

W

-LSB-

A

-RSB-

V

-LRB-

G

-RRB-

n

V

-LSB-

A

-RSB-

-LRB-

-RRB-

the

last

edge

ek

of

P

belongs

to

A

ek

A

so

that

k

is

an

odd

integer

and

ei

A

for

every

even

integer

i

k

-LRB-

-RRB-

every

edge

in

A

E

-LRB-

P

-RRB-

belongs

to

either

a

big

star

or

a

small

star

of

A

-LRB-

-RRB-

if

ei

-LRB-

xi

xi

-RRB-

An

-LRB-

i.e.

ei

is

an

edge

of

a

big

star

say

Si

of

A

-RRB-

then

xi

is

the

head

xi

is

a

tail

of

Si

and

xi

is

adjacent

to

no

tail

of

Si

in

G

xi

h

-LRB-

Si

-RRB-

xi

T

-LRB-

Si

-RRB-

-LRB-

xi

t

-RRB-

E

-LRB-

G

-RRB-

for

t

T

-LRB-

Si

-RRB-

and

-LRB-

-RRB-

if

ei

-LRB-

xi

xi

-RRB-

-LRB-

i.e.

ei

is

the

edge

of

a

small

star

S

of

A

-RRB-

then

-LRB-

xi

xi

-RRB-

-LRB-

as

well

as

ei

-LRB-

xi

xi

-RRB-

-RRB-

is

an

edge

of

G

An

example

of

a

passive

A

-LCB-

path

n

and

its

alternation

are

shown

in

Figure

In

this

and

every

figiure

below

a

thin

line

is

an

adge

in

E

-LRB-

G

-RRB-

n

A

a

thick

line

is

an

edge

in

A

and

a

broken

line

is

a

non

-LCB-

edge

of

G.

Figure

An

example

of

a

passive

A

-LCB-

path

n

and

its

alternation

A

path

consisting

of

one

vertex

in

W

-LSB-

A

-RSB-

is

a

trivial

passive

A

-LCB-

path

It

is

easy

to

see

that

Let

A

be

an

induced

n

-LCB-

packing

of

G

and

xk

xk

k

be

a

passive

A

-LCB-

path

Then

-LRB-

-RRB-

if

i

is

odd

-LRB-

i.e.

ei

A

-RRB-

then

the

subpath

of

is

a

passive

A

-LCB-

path

-LRB-

-RRB-

for

every

vertex

z

of

a

passive

path

xPy

there

exists

a

subpath

of

such

that

is

a

passive

A

-LCB-

path

and

z

is

an

end

-LCB-

vertex

of

the

last

edge

ei

-LRB-

xi

xi

-RRB-

of

z

fxi

xi

g

-LRB-

-RRB-

if

xi

is

the

head

of

a

big

star

of

A

then

i

is

even

-LRB-

-RRB-

if

i

is

even

and

xi

is

not

the

head

of

a

big

star

-LRB-

i.e.

-LRB-

xi

xi

-RRB-

form

a

small

star

of

A

and

-LRB-

xi

xi

-RRB-

E

-LRB-

G

-RRB-

n

A

-RRB-

then

the

path

xi

xi

is

a

passive

A

-LCB-

path

Clearly

passive

A

-LCB-

paths

have

the

following

important

exchange

property

Let

A

be

an

induced

n

-LCB-

packing

of

G

and

let

xPy

be

a

passive

A

-LCB-

path

Then

-LRB-

-RRB-

-LRB-

P

-RRB-

is

an

induced

n

-LCB-

packing

of

G

and

V

-LSB-

-RSB-

V

-LSB-

A

-RSB-

ny

-LSB-

x

so

that

jV

-LSB-

-RSB-

j

jV

-LSB-

A

-RSB-

j

and

xPy

is

not

an

augmenting

A

-LCB-

path

and

-LRB-

-RRB-

yPx

is

a

passive

-LCB-

path

In

other

words

a

passive

A

-LCB-

path

P

allows

us

to

move

from

a

given

induced

n

-LCB-

packing

A

to

another

induced

n

-LCB-

packing

whose

vertex

set

V

-LSB-

-RSB-

is

obtained

from

the

vertex

set

V

-LSB-

A

-RSB-

by

substituting

the

last

vertex

of

P

by

the

first

vertex

of

P

-LRB-

see

Figure

-RRB-

Note

that

the

requirment

-LRB-

xi

t

-RRB-

E

-LRB-

G

-RRB-

for

t

T

-LRB-

Si

-RRB-

in

-LRB-

-RRB-

is

essential

For

if

-LRB-

xi

t

-RRB-

is

an

edge

for

some

t

T

-LRB-

Si

-RRB-

then

-LRB-

P

-RRB-

is

not

an

induced

n

-LCB-

packing

namely

the

component

of

G

-LSB-

-RSB-

containing

Si

n

ei

is

not

an

induced

star

-LRB-

and

Figure

A

-LCB-

active

A

-LCB-

path

and

its

alternation

also

in

this

case

the

subpath

of

P

can

be

extended

to

the

augmenting

path

t

h

-LRB-

Si

-RRB-

-RRB-

Now

we

will

consider

some

special

augmenting

A

-LCB-

trails

-LRB-

so

called

active

A

-LCB-

trails

-RRB-

An

A

-LCB-

trail

P

xk

k

is

called

a

-LCB-

active

A

-LCB-

path

if

-LRB-

-RRB-

P

is

a

path

-LRB-

-RRB-

the

subpath

of

P

is

a

passive

A

-LCB-

path

-LRB-

and

so

W

-LSB-

A

-RSB-

-RRB-

and

-LRB-

-RRB-

xk

W

-LSB-

A

-RSB-

-LRB-

see

Figure

-RRB-

An

A

-LCB-

trail

P

xk

k

is

called

a

-LCB-

active

A

-LCB-

path

if

-LRB-

-RRB-

P

is

a

path

-LRB-

-RRB-

the

subpath

of

P

is

a

passive

A

-LCB-

path

-LRB-

and

so

W

-LSB-

A

-RSB-

-RRB-

and

-LRB-

-RRB-

xk

is

the

head

of

an

intermediate

star

say

S

and

xk

is

adjacent

to

no

tail

of

S

in

G

-LRB-

see

Figure

-RRB-

An

A

-LCB-

trail

P

xk

k

is

called

a

-LCB-

active

A

-LCB-

path

if

-LRB-

-RRB-

P

is

a

path

-LRB-

-RRB-

the

subpath

of

P

is

a

passive

A

-LCB-

path

-LRB-

and

so

W

-LSB-

A

-RSB-

-RRB-

and

-LRB-

-RRB-

xk

is

a

vertex

of

a

small

star

say

-LRB-

xk

y

-RRB-

and

-LRB-

xk

y

-RRB-

is

not

an

edge

of

G

-LRB-

see

Figure

-RRB-

An

A

-LCB-

trail

P

xk

k

is

called

a

-LCB-

active

A

-LCB-

path

if

-LRB-

-RRB-

P

is

a

path

-LRB-

-RRB-

the

subpath

of

P

is

a

passive

A

-LCB-

path

-LRB-

and

so

W

-LSB-

A

-RSB-

-RRB-

and

-LRB-

-RRB-

xk

and

xk

are

a

tail

and

the

head

of

some

non

-LCB-

small

star

Z

of

A

respectively

i.e.

xk

T

-LRB-

Z

-RRB-

and

xk

h

-LRB-

Z

-RRB-

-LRB-

see

Figure

-RRB-

Figure

A

-LCB-

active

A

-LCB-

path

and

its

alternation

Figure

A

-LCB-

active

A

-LCB-

path

and

its

alternation

Figure

A

-LCB-

active

A

-LCB-

path

and

its

alternation

An

active

A

-LCB-

path

is

an

s-active

A

-LCB-

path

for

some

s

In

other

words

the

only

way

to

create

an

active

A

-LCB-

path

in

G

is

to

add

to

a

passive

A

-LCB-

path

Q

a

new

edge

e

of

G

connecting

the

last

vertex

l

of

P

either

with

a

vertex

in

W

-LSB-

A

-RSB-

distinct

from

the

first

vertex

of

P

or

with

the

head

of

an

intermediate

star

of

A

or

with

a

vertex

of

a

small

star

Z

of

A

if

e

is

the

only

edge

between

l

and

Z

in

G

or

with

a

tail

of

a

non-small

star

of

A

In

particular

an

edge

with

both

end

-LCB-

vertices

in

W

-LSB-

A

-RSB-

is

a

active

A

-LCB-

path

we

call

such

path

a

trivial

active

A

-LCB-

path

A

trail

is

called

a

quasi

-LCB-

path

if

P

is

a

path

and

y

V

-LRB-

P

-RRB-

An

A

-LCB-

trail

xk

k

is

a

-LCB-

active

A

-LCB-

quasi-path

if

-LRB-

-RRB-

the

subtrail

is

a

passive

A

-LCB-

path

-LRB-

and

so

W

-LSB-

A

-RSB-

-RRB-

-LRB-

-RRB-

xk

xk

-LRB-

-RRB-

-LRB-

xk

xk

-RRB-

E

-LRB-

G

-RRB-

-LRB-

-RRB-

ek

-LRB-

xk

xk

-RRB-

and

ek

-LRB-

xk

xk

-RRB-

form

small

stars

of

A

and

-LRB-

-RRB-

the

-LCB-

vertex

set

fxk

xk

xk

xk

xk

induces

a

complete

subgraph

say

K

in

G

and

so

E

-LRB-

K

-RRB-

n

fek

ek

E

-LRB-

G

-RRB-

n

A

-LRB-

see

Figure

-RRB-

An

A

-LCB-

trail

xk

k

is

a

A

-LCB-

quasi-path

if

-LRB-

-RRB-

the

subtrail

is

a

passive

A

-LCB-

path

-LRB-

and

so

W

-LSB-

A

-RSB-

-RRB-

-LRB-

-RRB-

ek

-LRB-

xk

xk

-RRB-

is

an

edge

of

a

big

star

say

S

of

A

so

that

xk

h

-LRB-

S

-RRB-

and

xk

T

-LRB-

S

-RRB-

-LRB-

-RRB-

xk

xk

so

that

xk

h

-LRB-

S

-RRB-

xk

T

-LRB-

S

-RRB-

n

xk

and

-LRB-

xk

xk

-RRB-

E

-LRB-

G

-RRB-

-LRB-

-RRB-

the

edge

ek

-LRB-

xk

xk

-RRB-

forms

a

small

star

of

A

-LRB-

-RRB-

-LRB-

xk

xk

-RRB-

-LRB-

xk

xk

-RRB-

E

-LRB-

G

-RRB-

n

A

-LRB-

see

Figure

-RRB-

An

active

A

-LCB-

quasi-path

is

an

s-active

A

-LCB-

quasi-path

for

some

s

Clearly

an

active

A

-LCB-

quasi-path

is

a

quasi-path

Figure

A

-LCB-

active

A

-LCB-

quasi-path

and

its

alternation

Figure

A

-LCB-

active

A

-LCB-

quasi-path

and

its

alternation

An

active

A

-LCB-

trail

is

an

active

A

-LCB-

path

or

an

active

A

-LCB-

quasi-path

It

is

easy

to

see

that

active

A

-LCB-

trails

have

the

following

important

properties

Let

A

be

an

induced

n

-LCB-

packing

of

G

Then

-LRB-

-RRB-

if

xPy

is

a

passive

A

-LCB-

path

then

xQz

where

Q

P

n

y

is

an

active

Ay

-LCB-

path

in

G

n

y

where

Ay

A

E

-LRB-

G

n

y

-RRB-

and

-LRB-

-RRB-

if

xPy

is

an

active

A

-LCB-

path

or

an

active

A

-LCB-

quasi-path

then

-LRB-

P

-RRB-

is

an

induced

n

-LCB-

packing

of

G

and

V

-LSB-

A

-RSB-

ae

V

-LSB-

-RSB-

i.e.

an

active

A

-LCB-

path

is

an

augmenting

A

-LCB-

path

and

an

active

A

-LCB-

quasi-path

is

an

augmenting

A

-LCB-

quasi-path

Induced

n

-LCB-

star

packings

without

augmenting

trails

We

say

that

an

induced

n

-LCB-

packing

A

of

G

is

vertex

maximal

or

simply

V

-LCB-

maximal

if

there

is

no

induced

n

-LCB-

packing

of

G

such

that

V

-LSB-

A

-RSB-

ae

V

-LSB-

-RSB-

From

we

have

Let

A

be

a

V

-LCB-

maximal

induced

n

-LCB-

packing

of

G

Then

-LRB-

-RRB-

there

is

no

augmenting

A

-LCB-

trail

in

particular

-LRB-

-RRB-

there

is

no

active

A

-LCB-

path

and

no

active

A

-LCB-

quasi-path

in

G

and

in

particular

-LRB-

-RRB-

there

is

no

edge

having

both

its

end-vertices

in

W

-LSB-

A

-RSB-

We

will

use

the

following

notation

BA

is

the

set

of

vertices

of

G

that

can

be

reached

from

W

-LSB-

A

-RSB-

by

a

passive

A

-LCB-

path

BA

is

the

subgraph

of

G

induced

by

BA

-LRB-

i.e.

BA

G

-LSB-

BA

-RSB-

-RRB-

SA

is

the

set

of

big

stars

of

A

in

BA

TA

SfT

-LRB-

S

-RRB-

S

SAg

-LSB-

W

-LSB-

A

-RSB-

-RRB-

HA

is

the

set

of

heads

of

big

stars

in

BA

-LRB-

i.e.

HA

fh

-LRB-

S

-RRB-

S

SAg

-RRB-

CA

BA

n

HA

DA

V

-LRB-

G

-RRB-

n

BA

DA

G

-LSB-

DA

-RSB-

CA

BA

n

HA

and

C

A

G

-LSB-

CA

-RSB-

BA

n

HA

Let

A

be

an

induced

n

-LCB-

packing

of

G.

Let

S

be

a

star

of

A

Then

-LRB-

-RRB-

if

S

is

an

intermediate

star

then

V

-LRB-

S

-RRB-

BA

-LRB-

and

so

S

BA

;-RRB-

and

-LRB-

-RRB-

if

V

-LRB-

S

-RRB-

BA

then

S

BA

Proof

-LRB-

-RRB-

Let

S

be

an

an

intermediate

star

of

A.

Suppose

on

the

contrary

that

Z

V

-LRB-

S

-RRB-

BA

Then

for

every

vertex

x

in

Z

there

exists

a

passive

A

-LCB-

path

containing

x.

Let

P

be

a

shortest

passive

A

-LCB-

path

containing

a

vertex

in

Z

Then

the

last

edge

of

P

is

an

edge

of

S.

By

definition

of

a

passive

A

-LCB-

path

an

edge

of

the

passive

A

-LCB-

path

P

can

not

belong

to

the

intermediate

star

S

a

contradiction

-LRB-

-RRB-

Let

S

be

a

star

of

A

and

let

Z

V

-LRB-

S

-RRB-

BA

Let

z

Z.

By

-LRB-

-RRB-

S

is

not

an

intermediate

star

i.e.

S

is

either

a

small

star

or

a

big

star

of

A.

By

the

definition

of

BA

there

exists

a

passive

A

-LCB-

path

ek

containing

z

Then

contains

an

edge

say

e

-LRB-

xi

xi

-RRB-

of

S

incident

to

z

e

E

-LRB-

S

-RRB-

Therefore

e

is

a

common

edge

of

S

and

BA

Now

if

S

is

a

small

star

then

S

BA

and

we

are

done

Therefore

let

S

be

a

big

star

Then

h

-LRB-

S

-RRB-

xi

Since

is

a

passive

A

-LCB-

path

it

follows

from

the

definition

of

a

passive

A

-LCB-

path

that

for

every

tail

t

of

S

the

path

t

is

a

passive

A

-LCB-

path

Therefore

S

BA

From

we

have

Let

A

be

an

induced

n

-LCB-

packing

of

G

Then

A

DA

covers

all

the

vertices

of

DA

and

every

star

of

A

DA

is

a

star

of

A.

From

-LRB-

-RRB-

and

-LRB-

-RRB-

we

have

Let

A

be

an

induced

n

-LCB-

packing

of

G

Then

CA

is

the

set

of

the

last

vertices

of

all

passive

A

-LCB-

paths

of

G.

Let

A

be

an

induced

n

-LCB-

packing

of

G.

Suppose

that

-LRB-

-RRB-

P

xk

is

an

A

-LCB-

path

-LRB-

-RRB-

W

-LSB-

A

-RSB-

-LRB-

-RRB-

P

has

an

even

number

of

edges

so

that

-LRB-

xk

xk

-RRB-

A

-LRB-

-RRB-

P

is

not

a

passive

A

-LCB-

path

Then

G

has

an

active

A

-LCB-

path

starting

at

Proof

Since

is

a

-LRB-

trivial

-RRB-

passive

A

-LCB-

path

and

P

the

path

has

passive

A

-LCB-

subpaths

Let

be

a

-LRB-

the

-RRB-

maximal

passive

A

-LCB-

subpath

of

-LRB-

so

that

-LRB-

xm

xm

-RRB-

A

-RRB-

Since

is

not

a

passive

A

-LCB-

path

we

have

m

k

Let

us

consider

the

subpath

xm

xm

Clearly

em

A.

Suppose

that

the

edge

-LRB-

xm

xm

-RRB-

form

a

small

star

of

A

If

-LRB-

xm

xm

-RRB-

E

-LRB-

G

-RRB-

then

is

a

passive

A

-LCB-

path

of

P

and

ae

Therefore

the

passive

subpath

of

P

is

not

maximal

a

contradiction

If

-LRB-

xm

xm

-RRB-

E

-LRB-

G

-RRB-

then

is

a

-LCB-

active

A

-LCB-

path

Now

suppose

that

-LRB-

xm

xm

-RRB-

is

an

edge

of

a

non-small

star

S

of

A

If

xm

is

a

tail

of

S

then

is

a

-LCB-

active

A

-LCB-

path

Therefore

let

xm

be

the

head

of

S

If

-LRB-

xm

t

-RRB-

E

-LRB-

G

-RRB-

for

some

t

T

-LRB-

S

-RRB-

then

t

xm

is

a

-LCB-

active

A

-LCB-

path

Therefore

let

-LRB-

xm

t

-RRB-

E

-LRB-

G

-RRB-

for

every

t

T

-LRB-

S

-RRB-

If

S

is

an

intermediate

star

then

is

a

-LCB-

active

A

-LCB-

path

If

S

is

a

big

star

then

is

a

passive

A

-LCB-

subpath

of

P

containing

properly

Therefore

the

passive

A

-LCB-

subpath

of

P

is

not

maximal

a

contradiction

Let

A

be

an

induced

n

-LCB-

packing

of

G.

Suppose

that

-LRB-

-RRB-

xk

is

an

A

-LCB-

path

and

-LRB-

-RRB-

xk

W

-LSB-

A

-RSB-

Then

G

has

an

active

A

-LCB-

path

starting

at

xi

i

k.

Proof

Let

-LRB-

as

in

the

previous

proof

-RRB-

be

a

-LRB-

the

-RRB-

maximal

passive

A

-LCB-

subpath

of

-LRB-

so

that

-LRB-

xm

xm

-RRB-

A

-RRB-

If

m

k

then

by

G

has

an

active

A

-LCB-

path

starting

at

Therefore

let

m

k

and

so

is

a

passive

A

-LCB-

path

Since

xk

W

-LSB-

A

-RSB-

clearly

is

an

-LCB-

active

A

-LCB-

path

Since

is

also

an

A

-LCB-

path

the

same

is

true

for

xk

Let

BA

be

obtained

from

BA

by

deleting

the

edges

connecting

vertices

in

HA

i.e.

BA

BA

n

E

-LRB-

G

-LSB-

HA

-RSB-

-RRB-

From

we

have

Let

A

be

an

induced

n

-LCB-

packing

of

G.

Suppose

that

G

has

no

active

A

-LCB-

path

Then

BA

has

exactly

jW

-LSB-

A

-RSB-

j

components

each

containing

one

vertex

from

W

-LSB-

A

-RSB-

From

the

proofs

of

and

we

have

Let

the

assumption

of

either

or

holds

Then

G

has

an

active

A

-LCB-

path

R

such

that

-LRB-

-RRB-

R

contains

the

maximal

passive

A

-LCB-

path

P

of

P

and

moreover

-LRB-

-RRB-

R

is

obtained

from

P

by

one

of

the

four

operations

described

in

the

definition

of

an

active

A

-LCB-

path

so

that

either

R

P

or

R

G

-LSB-

P

-LSB-

t

-RSB-

where

t

is

a

tail

of

a

big

star

whose

head

is

the

vertex

of

P

adjacent

to

the

last

vertex

of

P

A

-LCB-

subpath

xPy

is

a

subpath

of

a

passive

A

-LCB-

path

such

that

-LRB-

-RRB-

every

odd

edge

is

in

E

-LRB-

G

-RRB-

n

A

in

particular

the

first

edge

-LRB-

x

-RRB-

is

in

E

-LRB-

G

-RRB-

n

A

-LRB-

-RRB-

every

even

edge

is

in

A

and

forms

a

small

star

of

A

and

-LRB-

-RRB-

the

last

edge

is

in

A

-LRB-

and

so

xPy

has

an

even

number

of

edges

-RRB-

Since

P

is

a

subpath

of

a

passive

A

-LCB-

path

every

small

star

G

-LSB-

xi

xi

-RSB-

in

P

is

connected

by

two

edges

with

the

previous

vertex

of

P

-LRB-

xi

xi

-RRB-

-LRB-

xi

xi

-RRB-

E

-LRB-

G

-RRB-

Let

A

be

an

induced

n

-LCB-

packing

of

G.

Suppose

that

-LRB-

-RRB-

P

xk

is

a

passive

A

-LCB-

path

-LRB-

so

that

W

-LSB-

A

-RSB-

-RRB-

-LRB-

-RRB-

xk

is

the

head

of

a

big

star

S

xk

h

-LRB-

S

-RRB-

and

xk

T

-LRB-

S

-RRB-

-LRB-

-RRB-

is

an

-LCB-

subpath

and

T

-LRB-

S

-RRB-

and

-LRB-

-RRB-

Q

Then

G

-LSB-

P

-LSB-

Q

-RSB-

has

an

active

A

-LCB-

trail

R

of

G

containing

P

n

xk

and

R

is

is

either

a

-LCB-

active

A

-LCB-

path

or

s

-LCB-

active

A

-LCB-

quasi-path

s

Proof

Let

-LRB-

-RRB-

ys

-LRB-

ys

-RRB-

Then

-LRB-

-RRB-

-LRB-

ys

ys

-RRB-

E

-LRB-

G

-RRB-

Put

h

h

-LRB-

S

-RRB-

xk

Since

is

a

passive

A

-LCB-

path

also

t

is

a

passive

A

-LCB-

path

for

every

t

T

-LRB-

S

-RRB-

In

particular

ti

is

a

passive

A

-LCB-

path

Let

be

a

-LRB-

the

-RRB-

maximal

passive

A

-LCB-

subpath

of

Clearly

Let

If

-LSB-

Q

n

i.e.

m

s

then

clearly

ym

ym

is

an

-LCB-

active

A

-LCB-

path

in

G

-LSB-

P

-LSB-

Q

-RSB-

containing

P

n

xk

Therefore

we

can

assume

that

-LSB-

Qn

By

the

same

reason

we

can

assume

that

-LSB-

Q

n

is

the

maximal

passive

A

-LCB-

subpath

of

the

A

-LCB-

path

-LSB-

Q.

Thus

Q

n

ti

is

a

-LCB-

subpath

for

i

Let

us

prove

that

if

-LRB-

yi

-RRB-

E

-LRB-

G

-RRB-

for

some

i

s

then

G

-LSB-

P

-LSB-

Q

-RSB-

has

a

-LCB-

active

A

-LCB-

quasi

-LCB-

path

containing

P

Let

k

be

the

minimum

integer

such

that

-LRB-

yk

-RRB-

E

-LRB-

G

-RRB-

Suppose

that

k

is

even

Then

-LRB-

yk

yk

-RRB-

and

-LRB-

yk

yk

-RRB-

and

also

-LRB-

yk

-RRB-

-LRB-

yk

E

-LRB-

G

-RRB-

n

A.

Clearly

-LRB-

yk

yk

-RRB-

E

-LRB-

G

-RRB-

n

A

Since

Q

n

is

an

-LCB-

subpath

starting

at

we

have

-LRB-

yk

yk

-RRB-

E

-LRB-

G

-RRB-

n

A

Since

Q

n

is

an

-LCB-

subpath

starting

at

we

have

-LRB-

yk

yk

-RRB-

E

-LRB-

G

-RRB-

n

A

If

-LRB-

yk

yk

-RRB-

E

-LRB-

G

-RRB-

then

yk

yk

yk

yk

is

an

-LCB-

active

A

-LCB-

path

in

G

-LSB-

P

-LSB-

Q

-RSB-

containing

P

nxk

Therefore

we

can

assume

that

-LRB-

yk

yk

-RRB-

E

-LRB-

G

-RRB-

Thus

G

-LSB-

yk

yk

-RSB-

is

a

complete

graph

Let

L

yk

yk

yk

yk

yk

Then

P

-LSB-

L

is

an

-LCB-

active

A

-LCB-

quasi-path

in

G

-LSB-

P

-LSB-

Q

-RSB-

containing

P

n

xk

If

k

is

odd

then

similar

arguments

show

that

G

has

a

required

active

A

-LCB-

trail

Thus

we

can

assume

that

-LRB-

yi

-RRB-

E

-LRB-

G

-RRB-

for

every

i

s

and

in

particular

for

i

s

s

Let

L

-LRB-

-RRB-

ys

ys

-LRB-

ys

-RRB-

Then

-LSB-

L

is

a

-LCB-

active

A

-LCB-

path

in

G

-LSB-

P

-LSB-

Q

-RSB-

containing

P

n

xk

Let

A

be

an

induced

n

-LCB-

packing

of

G.

Suppose

that

-LRB-

-RRB-

and

are

two

-LCB-

subpaths

-LRB-

-RRB-

and

and

-LRB-

-RRB-

either

or

and

there

exists

an

edge

of

G

connecting

a

vertex

of

with

a

vertex

of

Then

G

-LSB-

-LSB-

-RSB-

has

an

A

-LCB-

path

connecting

the

vertices

and

Proof

-LRB-

-RRB-

Suppose

that

and

e

-LRB-

-RRB-

E

-LRB-

G

-RRB-

where

V

-LRB-

-RRB-

and

V

-LRB-

-RRB-

Since

uiLivi

is

a

passive

-LCB-

subpath

G

-LSB-

Li

-RSB-

has

a

-LCB-

subpath

uiQzi

with

the

end

-LCB-

vertex

zi

i

Then

is

an

A

-LCB-

path

in

G

-LSB-

-LSB-

-RSB-

connecting

and

-LRB-

-RRB-

Suppose

that

Let

x

be

the

first

vertex

of

that

belongs

to

-LRB-

if

we

walk

along

from

-RRB-

Let

Li

-LRB-

yi

x

-RRB-

be

the

edge

of

Li

which

is

incident

to

x

and

which

does

not

belong

to

Lj

i

j

fi

jg

Clearly

at

most

one

of

the

edges

belongs

to

A

-LRB-

and

therefore

to

-RRB-

So

let

Since

is

a

-LCB-

subpath

G

-LSB-

-RSB-

has

a

-LCB-

subpath

Since

G

-LSB-

-RSB-

x

clearly

is

an

A

-LCB-

path

in

G

-LSB-

-LSB-

-RSB-

connecting

and

Let

A

be

an

induced

n

-LCB-

packing

of

G.

Let

xRy

and

be

two

maximal

-LCB-

subpaths

in

G.

Suppose

that

-LRB-

-RRB-

G

has

no

active

A

-LCB-

trail

and

-LRB-

-RRB-

x

Then

R

and

there

is

no

edge

of

G

connecting

R

with

Proof

-LRB-

uses

and

-RRB-

Let

xRy

be

a

maximal

-LCB-

subpath

in

G.

Clearly

x

TA

Let

R

denote

the

union

of

all

maximal

passive

-LCB-

subpaths

with

x.

Clearly

it

is

sufficient

to

prove

that

R

R

and

there

is

no

edge

in

G

connecting

R

with

R.

Suppose

the

contrary

Let

r

be

the

first

vertex

of

R

that

does

not

belong

to

R

and

is

linked

with

a

vertex

say

in

R

by

an

edge

say

e

-LRB-

r

-RRB-

Since

x

R

such

vertex

r

exists

Let

be

a

maximal

-LCB-

subpath

in

G

such

that

and

r.

Clearly

TA

If

e

-LRB-

r

-RRB-

A

then

e

and

so

r

But

r

a

contradiction

Therefore

e

A

Since

is

a

passive

A

-LCB-

subpath

there

exists

a

passive

A

-LCB-

path

wP

containing

By

the

property

of

the

vertex

r

we

have

xRr

-LRB-

P

-LSB-

-RRB-

and

in

particular

xRr

By

G

-LSB-

R

-LSB-

-RSB-

has

an

-LCB-

subpath

Since

G

-LSB-

R

-LSB-

-RSB-

P

clearly

Q

P

Therefore

wP

is

an

A

-LCB-

path

starting

at

w

W

-LSB-

A

-RSB-

and

terminating

at

x

We

recall

that

x

TA

Suppose

that

x

W

-LSB-

A

-RSB-

Then

x

w.

By

G

has

an

active

A

-LCB-

path

starting

at

x

a

contradiction

Now

suppose

tha

x

is

a

tail

of

a

big

star

S

x

T

-LRB-

S

-RRB-

Let

h

h

-LRB-

S

-RRB-

and

f

-LRB-

h

x

-RRB-

Suppose

that

T

-LRB-

S

-RRB-

Then

wTxfh

wDh

is

an

A

-LCB-

path

Since

the

last

vertex

h

of

wDh

is

the

head

of

a

big

star

and

the

last

edge

of

wDh

is

an

edge

of

a

big

star

D

is

not

a

passive

A

-LCB-

path

Since

w

W

-LSB-

A

-RSB-

the

A

-LCB-

path

wDh

satisfies

the

hypothesis

of

Therefore

by

G

has

an

active

A

-LCB-

path

starting

at

w

a

contradiction

Now

suppose

that

T

-LRB-

S

-RRB-

We

recall

that

x

T

-LRB-

S

-RRB-

and

x

Then

by

G

has

either

an

active

A

-LCB-

path

or

an

active

A-quasi

-LCB-

path

a

contradiction

An

-LCB-

cycle

Q

of

G

is

a

cycle

obtained

from

an

-LCB-

subpath

say

xLy

of

G

by

adding

the

edge

of

G

connecting

the

end

-LCB-

vertices

of

the

-LCB-

subpath

L

Q

L

-LSB-

e

where

e

-LRB-

x

y

-RRB-

E

-LRB-

G

-RRB-

It

is

clear

that

If

an

-LCB-

cycle

has

an

even

number

of

vertices

then

Q

is

a

perfect

matching

of

Q

If

an

-LCB-

cycle

has

an

odd

number

of

vertices

then

Q

is

a

perfect

matching

of

Q

n

z

for

some

vertex

z

of

Q.

Let

A

be

an

induced

n

-LCB-

packing

of

G.

Suppose

that

-LRB-

-RRB-

xk

is

a

passive

A

-LCB-

path

-LRB-

so

that

W

-LSB-

A

-RSB-

-RRB-

-LRB-

-RRB-

Q

is

an

-LCB-

cycle

and

-LRB-

-RRB-

one

of

the

following

holds

-LRB-

-RRB-

Q

has

an

odd

number

of

edges

and

P

Q

xk

and

-LRB-

-RRB-

Q

has

an

even

number

of

edges

P

Q

and

-LRB-

xk

t

-RRB-

E

-LRB-

G

-RRB-

where

t

V

-LRB-

Q

-RRB-

Then

one

of

the

following

conditions

holds

-LRB-

-RRB-

G

-LSB-

P

-LSB-

Q

-RSB-

has

an

active

A

-LCB-

trail

containing

P

and

this

A

-LCB-

trail

is

either

a

-LCB-

active

A

-LCB-

path

or

a

-LCB-

active

A

-LCB-

quasi-path

and

-LRB-

-RRB-

G

-LSB-

Q

-LSB-

xk

-RSB-

is

a

complete

graph

Proof

-LRB-

uses

and

-RRB-

Let

Q

-LRB-

t

-RRB-

fk

-LRB-

yk

t

-RRB-

where

t

xk

in

case

-LRB-

h

-RRB-

We

prove

the

statement

for

case

-LRB-

-RRB-

The

proof

for

case

-LRB-

-RRB-

is

similar

-LRB-

-RRB-

Let

us

consider

an

A

-LCB-

path

P

-LSB-

Q

n

fk

Clearly

W

-LSB-

A

-RSB-

and

has

an

even

number

of

vertices

Suppose

that

is

not

a

passive

A

-LCB-

path

Then

by

G

has

an

active

A

-LCB-

path

Moreover

since

P

is

a

passive

A

-LCB-

path

and

Q

n

fk

is

an

-LCB-

subpath

it

follows

from

and

that

G

-LSB-

-RSB-

has

an

-LCB-

active

A

-LCB-

path

containing

P

Thus

we

can

assume

that

P

-LSB-

Q

n

fk

is

a

passive

A

-LCB-

path

Therefore

-LRB-

yi

yi

-RRB-

E

-LRB-

G

-RRB-

n

A

for

every

odd

i

k

Similarly

we

can

assume

that

P

-LSB-

Q

n

is

a

passive

A

-LCB-

path

Therefore

-LRB-

yi

yi

-RRB-

E

-LRB-

G

-RRB-

for

every

even

i

k

Suppose

that

-LRB-

yj

yj

-RRB-

E

-LRB-

G

-RRB-

for

some

even

j

k

We

can

assume

that

j

is

the

minimum

number

for

which

it

occures

Then

PtQyj

yj

yj

yj

yj

is

a

-LCB-

active

A

-LCB-

path

in

G

-LSB-

P

-LSB-

Q

-RSB-

containing

P

Now

suppose

that

-LRB-

yi

yi

-RRB-

E

-LRB-

G

-RRB-

for

every

even

i

k

Put

Di

G

-LSB-

yi

yi

-RSB-

for

every

even

i

k

By

the

above

arguments

Di

is

a

complete

graph

-LRB-

-RRB-

Let

us

prove

by

induction

on

s

Qj

the

following

Claim

If

G

-LSB-

P

-LSB-

Q

-RSB-

has

no

-LCB-

active

A

-LCB-

quasi-path

containing

P

then

G

-LSB-

Q

-RSB-

is

a

complete

graph

If

s

then

the

statement

follows

from

-LRB-

-RRB-

Let

s

If

-LRB-

t

-RRB-

E

-LRB-

G

-RRB-

then

put

t

If

-LRB-

t

-RRB-

E

-LRB-

G

-RRB-

then

put

t

Then

P

-LSB-

Li

i

is

an

-LCB-

active

A

-LCB-

quasi-path

containing

P

a

contradiction

Therefore

-LRB-

t

yi

-RRB-

E

-LRB-

G

-RRB-

n

A

for

i

By

similar

arguments

-LRB-

t

yi

-RRB-

E

-LRB-

G

-RRB-

for

i

k

k

Put

-LSB-

-LRB-

t

-RRB-

Qnfyk

yk

-LSB-

-LRB-

t

yk

-RRB-

and

-LSB-

-LRB-

-RRB-

Then

Qi

is

a

closed

-LCB-

subpath

Qi

P

t

and

Qij

s

for

i

By

the

inductive

hypothesis

G

-LSB-

Qi

-RSB-

is

a

complete

graph

for

i

Therefore

G

-LSB-

Q

-RSB-

is

a

complete

graph

Remark

The

above

statement

can

also

be

proved

by

induction

on

Qj

by

contracting

a

triangle

containing

xk

and

a

small

star

of

Q.

From

and

we

have

Let

A

be

an

induced

n

-LCB-

star

packing

in

G.

Suppose

that

-LRB-

-RRB-

G

has

no

active

A

-LCB-

trail

-LRB-

-RRB-

Q

is

an

-LCB-

cycle

and

-LRB-

-RRB-

there

exists

a

TA

such

that

either

Q

TA

and

-LRB-

a

q

-RRB-

E

-LRB-

G

-RRB-

or

Q

TA

a

for

some

q

V

-LRB-

Q

-RRB-

Then

K

G

-LSB-

Q

-LSB-

a

-RSB-

is

a

complete

graph

and

K

is

a

perfect

matching

of

K

Given

a

TA

let

Y

a

denote

the

union

of

all

-LCB-

subpaths

starting

at

a.

Let

G

-LSB-

Y

a

-RSB-

In

other

words

is

the

subgraph

of

G

induced

by

the

set

of

vertices

that

can

be

reached

from

the

vertex

a

by

a

-LCB-

subpath

We

call

an

-LCB-

subgraph

of

G

It

easy

to

see

that

Let

A

be

an

induced

n

-LCB-

packing

of

G

and

let

a

TA

Then

-LRB-

-RRB-

is

a

connected

subgraph

of

G

and

-LRB-

-RRB-

A

is

a

perfect

matching

of

-LRB-

and

so

has

an

odd

number

of

vertices

-RRB-

From

we

have

Let

A

be

an

induced

n

-LCB-

packing

of

G.

Let

a

b

TA

and

a

b.

Suppose

that

G

has

no

active

A

-LCB-

trail

Then

N

b

and

G

has

no

edge

connecting

a

vertex

in

with

a

vertex

in

N

b.

Let

Cmp

-LRB-

F

-RRB-

denote

the

set

of

connected

components

of

a

graph

F

By

the

definition

of

C

A

for

every

vertex

x

of

C

A

there

exists

a

TA

such

that

x

Therefore

from

and

we

have

Let

A

be

an

induced

n

-LCB-

packing

of

G.

Suppose

that

G

has

no

active

A

-LCB-

trail

Then

Cmp

-LRB-

C

A

-RRB-

fNa

a

TAg

Let

A

be

an

induced

n

-LCB-

packing

of

G.

Suppose

that

G

has

no

active

A

-LCB-

trail

Then

G

has

no

edge

connecting

a

vertex

in

CA

with

a

vertex

in

DA

Proof

-LRB-

uses

and

-RRB-

Suppose

on

the

contrary

that

there

exists

an

edge

-LRB-

c

d

-RRB-

of

G

such

that

c

CA

and

d

DA

-RRB-

By

c

is

the

end

-LCB-

vertex

of

a

passive

path

say

wP

c

and

so

w

W

-LSB-

A

-RSB-

and

c

is

either

a

tail

of

a

big

star

or

an

end

-LCB-

vertex

of

a

small

star

Therefore

-LRB-

c

d

-RRB-

E

-LRB-

G

-RRB-

By

every

vertex

of

DA

belongs

to

a

star

of

A

which

is

a

subgraph

of

DA

Let

-LRB-

d

-RRB-

be

the

edge

of

the

star

of

A

in

DA

containing

the

vertex

d

and

so

DA

Put

P

wP

c

d

If

P

is

a

passive

A

-LCB-

path

then

by

the

definition

of

CA

we

have

CA

a

contradiction

If

P

is

not

a

passive

A

-LCB-

path

then

by

G

has

an

active

A

-LCB-

path

a

contradiction

From

and

we

have

Let

A

be

an

induced

n

-LCB-

packing

and

let

a

TA

Suppose

that

G

has

no

active

A

-LCB-

trail

Then

is

a

connected

component

of

GnHA

i.e.

Cmp

-LRB-

C

A

-RRB-

Cmp

-LRB-

GnHA

-RRB-

Since

each

vertex

of

HA

is

adjacent

to

at

least

one

vertex

-LRB-

actually

at

least

two

vertices

-RRB-

of

CA

in

G

we

have

from

Let

A

be

an

induced

n

-LCB-

packing

and

let

a

TA

Suppose

that

G

has

no

active

A

-LCB-

trail

Then

HA

is

the

set

of

vertices

in

GnCA

adjacent

to

at

least

one

vertex

in

CA

We

now

can

describe

the

structure

of

a

subgraph

Let

A

be

an

induced

n

-LCB-

packing

and

let

a

TA

Suppose

that

G

has

no

active

A

-LCB-

trail

Then

is

an

odd

clique

tree

Proof

-LRB-

uses

-RRB-

Let

us

prove

the

statement

by

induction

on

the

number

of

vertices

of

If

has

one

vertex

i.e.

a

then

the

statement

is

obviously

true

So

let

jV

-LRB-

-RRB-

j

Since

jV

-LRB-

-RRB-

j

is

odd

clearly

jV

-LRB-

-RRB-

j

By

the

definition

of

there

exist

V

-LRB-

-RRB-

such

that

G

-LSB-

a

-RSB-

is

a

triangle

and

the

edge

e

-LRB-

-RRB-

forms

a

small

star

of

A.

Let

G

and

N

a

be

obtained

from

GnE

-LRB-

-RRB-

and

NanE

-LRB-

-RRB-

respectively

by

identifying

and

with

the

vertex

a.

Let

A

A

n

e

It

is

easy

to

see

that

A

is

an

induced

n

-LCB-

packing

in

G

a

T

A

-LRB-

G

-RRB-

and

N

a

is

an

-LCB-

subgraph

of

G

It

is

also

clear

that

every

active

A

-LCB-

trail

in

G

can

be

easily

transformed

into

an

active

A

-LCB-

trail

in

G

Since

G

has

no

active

A

-LCB-

trail

G

also

has

no

active

A

-LCB-

trail

Clearly

jV

-LRB-

N

a

-RRB-

j

jV

-LRB-

-RRB-

j

Therefore

by

the

inductive

hypothesis

N

a

is

an

odd

clique

tree

Let

R

be

a

block

of

N

a

that

does

not

contain

a

Then

clearly

R

is

also

a

block

of

Since

N

a

is

an

odd

clique

tree

R

is

an

odd

complete

graph

Let

C

be

a

block

of

N

a

containing

a

Then

every

vertex

q

of

C

n

a

is

adjacent

to

a

vertex

of

in

G.

Suppose

that

there

exists

a

vertex

z

of

such

that

-LRB-

z

q

-RRB-

E

-LRB-

G

-RRB-

and

-LRB-

t

q

-RRB-

E

-LRB-

G

-RRB-

for

every

q

V

-LRB-

C

n

a

-RRB-

and

every

t

V

-LRB-

-RRB-

n

z

Then

C

G

-LSB-

C

n

a

-LSB-

z

-RSB-

is

an

odd

complete

graph

and

is

a

block

of

In

particular

if

z

a

then

C

C

is

a

block

of

containing

a

If

z

a

then

C

is

a

block

of

avoiding

a

So

we

may

conclude

in

particular

that

every

block

of

avoiding

a

is

an

odd

complete

graph

Now

suppose

that

there

exist

two

distinct

vertices

and

of

and

two

distinct

vertices

and

of

C

n

a

such

that

-LRB-

-RRB-

E

-LRB-

G

-RRB-

and

-LRB-

-RRB-

E

-LRB-

G

-RRB-

Put

M

-LRB-

C

-RRB-

A

-LRB-

C

n

a

-RRB-

Clearly

M

-LRB-

C

-RRB-

is

a

perfect

matching

of

C

n

a.

Therefore

since

C

n

a

is

a

complete

graph

and

G

-LSB-

a

-RSB-

is

a

triangle

containing

a

small

star

G

-LSB-

-RSB-

the

subgraph

F

G

-LSB-

-LSB-

C

n

a

-RRB-

-RSB-

has

an

-LCB-

cycle

Q

such

that

either

V

-LRB-

Q

-RRB-

V

-LRB-

C

-RRB-

and

Q

TA

a

or

V

-LRB-

Q

-RRB-

V

-LRB-

F

n

a

-RRB-

and

-LRB-

a

xi

-RRB-

E

-LRB-

G

-RRB-

Since

G

has

no

active

A

-LCB-

trail

by

C

is

a

complete

graph

and

M

-LRB-

C

-RRB-

is

a

perfect

matching

of

F

n

a.

Suppose

that

there

are

two

different

complete

subgraphs

and

of

G

such

that

ae

Fi

and

M

-LRB-

Fi

-RRB-

is

a

a

perfect

matching

of

Fina

i

Then

H

-LSB-

contains

an

-LCB-

cycle

L

such

that

either

V

-LRB-

L

-RRB-

V

-LRB-

H

-RRB-

and

L

TA

a

or

V

-LRB-

L

-RRB-

V

-LRB-

H

n

a

-RRB-

and

-LRB-

a

xi

-RRB-

E

-LRB-

G

-RRB-

Since

G

has

no

active

A

-LCB-

trail

by

H

is

a

complete

graph

and

M

-LRB-

H

-RRB-

is

a

perfect

matching

of

H

n

a.

Thus

every

block

of

containing

a

is

an

odd

complete

graph

Since

is

a

connected

graph

and

every

its

block

is

an

odd

complete

graph

is

an

odd

clique

tree

Let

Oct

-LRB-

F

-RRB-

denote

the

set

of

components

of

F

that

are

odd

clique

trees

From

and

we

have

Let

A

be

an

induced

n

-LCB-

packing

of

G.

Suppose

that

G

has

no

active

A

-LCB-

trail

Then

every

component

of

C

A

is

an

odd

clique

tree

i.e.

Cmp

-LRB-

C

A

-RRB-

Oct

-LRB-

C

A

-RRB-

Let

T

be

an

odd

clique

tree

Then

T

is

an

ISn

-LCB-

critical

graph

for

every

integer

n

-LRB-

and

in

particular

T

is

a

matching

-LCB-

critical

graph

-RRB-

Proof

-LRB-

-RRB-

Let

us

prove

that

T

T

nx

has

a

perfect

matching

for

every

x

V

-LRB-

G

-RRB-

Clearly

T

has

the

properties

-LRB-

P

-RRB-

every

block

of

T

is

a

clique

and

-LRB-

P

-RRB-

every

component

of

T

has

exactly

one

maximal

clique

with

an

even

number

of

vertices

Let

us

prove

by

induction

of

the

number

of

vertices

that

a

graph

with

properties

-LRB-

P

-RRB-

and

-LRB-

P

-RRB-

has

a

perfect

matching

Clearly

the

statement

holds

for

the

graph

with

one

edge

Let

T

be

obtained

from

T

by

deleting

an

arbitrary

pair

of

-LRB-

adjacent

-RRB-

vertices

Clearly

T

also

has

properties

-LRB-

P

-RRB-

and

-LRB-

P

-RRB-

By

the

inductive

hypothesis

T

has

a

perfect

matching

Therefore

T

also

has

a

perfect

matching

-LRB-

-RRB-

Now

let

us

prove

by

induction

on

the

number

of

vertices

that

T

has

no

perfect

induced

k

-LCB-

packing

for

every

integer

k

Clearly

the

statement

holds

for

the

trivial

graph

with

one

vertex

Suppose

that

T

has

a

perfect

induced

n

-LCB-

packing

A

for

some

integer

n

Let

S

be

a

star

of

A

and

let

T

T

n

V

-LRB-

S

-RRB-

Then

at

least

one

component

C

of

T

is

an

odd

clique

tree

By

the

inductive

hypothesis

T

has

no

perfect

induced

n

-LCB-

packing

for

every

integer

k

On

the

other

hand

A

-LSB-

C

is

a

perfect

induced

n

-LCB-

packing

of

C

a

contradiction

From

and

we

have

Let

A

be

an

induced

n

-LCB-

packing

and

let

a

TA

Suppose

that

G

has

no

active

A

-LCB-

trail

Then

Oct

-LRB-

GnHA

-RRB-

Oct

-LRB-

C

A

-RRB-

Cmp

-LRB-

C

A

-RRB-

fNa

a

TAg

Cmp

-LRB-

GnHA

-RRB-

Duality

theorem

for

the

induced

n

-LCB-

star

packings

in

a

graph

Now

we

are

ready

to

prove

the

Duality

Theorem

First

we

shall

establish

the

corresponding

lower

bound

on

the

number

of

vertices

jW

-LSB-

A

-RSB-

j

in

a

graph

G

that

are

not

covered

by

a

given

induced

n

-LCB-

packing

A.

Given

a

vertex

subset

Z

of

a

graph

F

let

OctZ

-LRB-

F

-RRB-

denote

the

set

of

all

components

C

of

F

such

that

C

is

an

odd

clique

tree

and

V

-LRB-

C

-RRB-

Z

If

in

particular

Z

V

-LRB-

F

-RRB-

then

OctZ

-LRB-

F

-RRB-

Oct

-LRB-

F

-RRB-

is

the

set

of

components

of

F

that

are

odd

clique

trees

Let

A

be

an

induced

n

-LCB-

packing

of

G

and

X

and

Z

be

arbitrary

vertex

subsets

of

G

Then

jW

-LSB-

A

-RSB-

Zj

jOctZ

-LRB-

G

n

X

-RRB-

j

njXj

and

in

particular

if

Z

V

-LRB-

G

-RRB-

jW

-LSB-

A

-RSB-

j

jOct

-LRB-

G

n

X

-RRB-

j

njXj

Proof

-LRB-

uses

-RRB-

Let

OZ

-LRB-

F

-RRB-

denote

the

union

of

all

components

that

are

members

of

OctZ

-LRB-

F

-RRB-

Let

be

the

subset

of

edges

of

A

that

are

not

incident

to

X.

Let

Q

be

the

set

of

vertices

in

OZ

-LRB-

GnX

-RRB-

that

are

not

covered

by

the

edge

set

By

an

odd

clique

tree

does

not

have

a

perfect

induced

n

-LCB-

packing

-LRB-

and

so

its

vertices

can

not

be

covered

by

an

induced

n

-LCB-

packing

of

G

-RRB-

Therefore

every

component

in

OctZ

-LRB-

G

n

X

-RRB-

has

at

least

one

vertex

that

is

not

covered

by

the

edge

set

of

i.e.

Q

C

for

every

component

-LRB-

odd

clique

tree

-RRB-

C

OctZ

-LRB-

G

n

X

-RRB-

Therefore

jQj

jOctZ

-LRB-

G

n

X

-RRB-

j

Since

every

star

in

the

induced

n

-LCB-

packing

A

has

at

most

n

edges

it

follows

that

at

most

njXj

vertices

in

Q

can

be

covered

by

the

edge

set

Therefore

at

least

jQj?njXj

jOctZ

-LRB-

GnX

-RRB-

j?njXj

vertices

remain

uncovered

by

A.

Since

jT

Aj

jW

-LSB-

A

-RSB-

j

njHAj

we

have

from

Let

A

be

an

induced

n

-LCB-

packing

in

G.

Suppose

that

G

has

no

active

A

-LCB-

trail

Then

jW

-LSB-

A

-RSB-

j

jOct

-LRB-

G

n

HA

-RRB-

j

njHAj

Now

we

can

easily

prove

the

Duality

Theorem

Let

n

Then

minfjW

-LSB-

B

-RSB-

j

B

PISn

-LRB-

G

-RRB-

g

maxfjOct

-LRB-

G

n

X

-RRB-

j

njXj

X

V

-LRB-

G

-RRB-

g

jOct

-LRB-

G

n

HA

-RRB-

j

njHAj

jW

-LSB-

A

-RSB-

j

for

every

V

-LCB-

maximal

-LRB-

maximum

-RRB-

induced

n

-LCB-

packing

A

in

G.

Proof

-LRB-

uses

and

-RRB-

By

jW

-LSB-

B

-RSB-

j

maxfjOct

-LRB-

G

n

X

-RRB-

j

njXj

X

V

-LRB-

G

-RRB-

g

for

every

induced

n

-LCB-

packing

B

of

G.

By

an

active

A

-LCB-

trail

is

an

augmenting

A

-LCB-

trail

Let

A

be

a

V

-LCB-

maximal

induced

n

-LCB-

packing

of

G

Then

G

has

no

augmenting

A

-LCB-

trail

-LRB-

see

-RRB-

and

therefore

has

no

active

A

-LCB-

trail

By

jW

-LSB-

A

-RSB-

j

jOct

-LRB-

GnHA

-RRB-

j?njHAj

Therefore

minfjW

-LSB-

B

-RSB-

j

B

PISn

-LRB-

G

-RRB-

g

jW

-LSB-

A

-RSB-

j

jOct

-LRB-

G

n

HA

-RRB-

j

njHAj

maxfjOct

-LRB-

G

n

X

-RRB-

j

njXj

X

V

-LRB-

G

-RRB-

g.

From

and

we

have

the

following

strengthening

of

Theorem

An

induced

n

-LCB-

packing

A

of

G

is

V

-LCB-

maximum

-LRB-

V

-LCB-

maximal

-RRB-

if

and

only

if

G

has

no

active

A

-LCB-

trail

Matroid

generated

by

the

induced

n

-LCB-

star

pack

ings

Let

ISn

-LRB-

G

-RRB-

denote

the

family

of

vertex

subsets

of

G

that

can

be

covered

by

an

induced

n

-LCB-

star

packing

of

G

In

this

section

we

will

show

that

ISn

-LRB-

G

-RRB-

is

the

independence

set

of

a

matroid

and

we

will

describe

the

structure

of

a

circuit

of

this

matroid

As

we

have

already

mentioned

before

ISn

has

the

hereditary

property

-LRB-

h

-RRB-

X

Y

ISn

-RRB-

X

ISn

Therefore

ISn

is

an

independence

family

of

subsets

of

V

-LRB-

G

-RRB-

Let

A

be

a

V

-LCB-

maximal

induced

n

-LCB-

star

packing

of

G

i.e.

V

-LSB-

A

-RSB-

is

a

maximal

set

-LRB-

a

base

-RRB-

in

ISn

We

shall

use

the

following

notation

similar

to

that

in

Section

BAz

is

the

set

of

vertices

in

G

that

can

be

reached

from

z

by

a

passive

A

-LCB-

path

BAz

is

the

subgraph

of

G

induced

by

BAz

i.e.

BAz

G

-LSB-

BAz

-RSB-

SAz

is

the

set

of

big

stars

of

A

in

BAz

TAz

is

the

set

of

tails

of

all

big

stars

in

BAz

plus

the

vertex

z

i.e.

TAz

SfT

-LRB-

S

-RRB-

S

Sz

-LRB-

A

-RRB-

g

-LSB-

fzg

HAz

is

the

set

of

heads

of

big

stars

in

BAz

i.e.

HAz

fh

-LRB-

S

-RRB-

S

Sz

-LRB-

A

-RRB-

g

CAz

BAz

n

HAz

Nz

-LRB-

A

-RRB-

fNa

a

TAz

g.

Clearly

jNz

-LRB-

A

-RRB-

j

jT

Az

j

njHAz

j

The

following

statement

on

CAz

is

similar

to

on

CA

Let

A

be

an

induced

n

-LCB-

packing

of

G

Then

CAz

is

the

set

of

the

last

vertices

of

passive

A

-LCB-

paths

of

G

starting

at

z

The

next

statement

is

similar

to

Let

A

be

a

V

-LCB-

maximal

induced

n

-LCB-

packing

of

G

and

z

W

-LSB-

A

-RSB-

Then

every

a

TAz

is

a

connected

component

of

G

n

HAz

Proof

-LRB-

uses

and

-RRB-

Put

Gz

G

n

HAz

Let

a

TAz

Obviously

Gz

Suppose

on

the

contrary

that

is

not

a

component

of

Gz

Then

there

exists

an

edge

-LRB-

x

y

-RRB-

of

G

such

that

x

V

-LRB-

-RRB-

and

y

V

-LRB-

Gz

-RRB-

n

V

-LRB-

-RRB-

By

y

HA

Therefore

y

HA

n

HAz

By

the

definitions

of

HA

and

HAz

there

exists

a

big

star

S

of

A

such

that

h

-LRB-

S

-RRB-

y

and

S

BAz

Since

x

and

a

TAz

clearly

x

CAz

By

there

exists

a

passive

A

-LCB-

path

P

of

G

starting

at

z

and

terminating

at

x

Then

P

Pxeyett

where

et

-LRB-

y

t

-RRB-

is

an

edge

of

S

and

t

is

a

tail

of

S

is

a

passive

A

-LCB-

path

from

z

to

t.

Therefore

et

E

-LRB-

BAz

-RRB-

for

every

t

T

-LRB-

S

-RRB-

and

so

S

BAz

a

contradiction

Let

A

be

a

V

-LCB-

maximal

induced

n

-LCB-

packing

of

G

and

z

W

-LSB-

A

-RSB-

Then

-LRB-

-RRB-

there

is

no

induced

n

-LCB-

packing

of

G

that

covers

CAz

and

-LRB-

-RRB-

for

every

x

CAz

there

exists

an

induced

n

-LCB-

packing

Ax

such

that

V

-LSB-

Ax

-RSB-

V

-LSB-

A

-RSB-

n

x

-LSB-

z.

Proof

-LRB-

uses

and

-RRB-

Let

us

prove

-LRB-

-RRB-

Put

CAz

Z

and

HAz

X.

Let

a

TAz

Obviously

V

-LRB-

-RRB-

Z.

By

is

a

component

of

G

n

X.

By

is

an

odd

clique

tree

Hence

jOctZ

-LRB-

GnX

-RRB-

j

jNz

-LRB-

A

-RRB-

j

jT

Az

j

njHAz

j

njXj

Therefore

by

jW

-LSB-

A

-RSB-

Zj

jOctZ

-LRB-

G

n

X

-RRB-

j

njXj

Thus

every

induced

n

-LCB-

packing

in

G

does

not

cover

at

least

one

vertex

in

CAz

Let

us

prove

-LRB-

-RRB-

Let

x

CAz

Then

by

there

exists

a

passive

A

-LCB-

path

zPx

of

G

from

z

to

x

Then

by

Ax

-LRB-

zPx

-RRB-

is

the

required

induced

n

-LCB-

packing

Now

we

can

describe

a

circuit

of

the

independence

set

ISn

Let

A

be

a

maximal

induced

n

-LCB-

packing

of

G

Then

-LRB-

-RRB-

CAz

is

a

circuit

of

the

independence

set

ISn

of

subsets

in

V

-LRB-

G

-RRB-

and

-LRB-

-RRB-

CAz

is

the

unique

circuit

of

ISn

in

the

vertex

set

V

-LSB-

A

-RSB-

-LSB-

z.

Proof

-LRB-

uses

-RRB-

Let

us

prove

-LRB-

-RRB-

By

-LRB-

-RRB-

CAz

ISn

By

-LRB-

-RRB-

X

ISn

for

every

proper

subset

X

ae

CAz

Therefore

CAz

is

a

circuit

of

ISn

Let

us

prove

-LRB-

-RRB-

Suppose

on

the

contrary

that

there

exists

a

subset

C

of

V

-LSB-

A

-RSB-

-LSB-

z

that

is

a

circuit

of

ISn

distinct

from

CAz

Since

a

circuit

is

a

minimal

subset

of

V

-LRB-

G

-RRB-

that

is

not

a

member

of

ISn

we

have

CAz

C.

Therefore

there

exists

x

CAz

n

C

Then

C

V

-LSB-

A

-RSB-

n

x

-LSB-

z.

By

-LRB-

-RRB-

there

exists

an

induced

n

-LCB-

packing

Ax

such

that

V

-LSB-

Ax

-RSB-

V

-LSB-

A

-RSB-

n

x

-LSB-

z.

Therefore

the

induced

n

-LCB-

packing

Ax

covers

C

and

so

C

ISn

a

contradiction

It

is

easy

to

prove

the

following

fact

for

matroids

-LSB-

-RSB-

Let

A

be

a

family

of

subsets

of

a

finite

set

E

having

the

hereditary

property

X

Y

A

-RRB-

X

A.

Suppose

that

for

every

maximal

subset

B

in

A

and

for

every

element

e

E

n

B

the

set

B

-LSB-

z

contains

the

unique

circuit

of

the

family

A

Then

A

is

the

independence

set

of

a

matroid

defined

on

E

Now

we

can

prove

Theorem

-LRB-

-RRB-

The

set

ISn

-LRB-

G

-RRB-

of

vertex

subsets

of

G

is

the

independence

set

of

a

matroid

Proof

-LRB-

uses

and

-RRB-

Let

B

be

a

maximal

subset

in

the

independence

set

ISn

-LRB-

G

-RRB-

of

subsets

in

V

-LRB-

G

-RRB-

and

let

z

V

-LRB-

G

-RRB-

n

B.

By

B

-LSB-

z

contains

the

unique

circuit

of

the

independence

set

ISn

-LRB-

G

-RRB-

Therefore

by

ISn

-LRB-

G

-RRB-

is

the

independence

set

of

a

matroid

defined

on

V

-LRB-

G

-RRB-

Let

MISn

-LRB-

G

-RRB-

denote

the

matroid

of

G

with

the

independence

set

ISn

-LRB-

G

-RRB-

i.e.

MISn

-LRB-

G

-RRB-

-LRB-

V

-LRB-

G

-RRB-

ISn

-LRB-

G

-RRB-

-RRB-

We

need

some

notions

and

notation

from

the

matroid

theory

-LSB-

-RSB-

M

is

a

matroid

with

the

ground

set

E

C

-LRB-

M

-RRB-

is

the

set

of

cyclic

elements

of

M

-LRB-

i.e.

the

elements

belonging

to

at

least

one

circuit

of

M

-RRB-

and

L

-LRB-

M

-RRB-

is

the

set

of

coloops

-LRB-

i.e.

one

element

cocircuits

-RRB-

of

M

Given

a

base

B

-LRB-

i.e.

a

maximal

independent

set

-RRB-

of

M

and

an

element

e

E

n

B

let

C

-LRB-

B

e

-RRB-

denote

the

circuit

of

M

that

is

a

subset

of

B

-LSB-

e

It

is

known

-LSB-

-RSB-

and

it

is

easy

to

prove

that

Let

M

be

a

matroid

Then

-LRB-

-RRB-

C

-LRB-

M

-RRB-

SfC

-LRB-

B

e

-RRB-

e

E

n

Bg

-LRB-

-RRB-

C

-LRB-

M

-RRB-

is

the

set

of

elements

that

do

not

belong

to

at

least

one

base

of

M

and

-LRB-

-RRB-

L

-LRB-

M

-RRB-

E

n

C

-LRB-

M

-RRB-

Let

A

be

a

maximal

induced

n

-LCB-

packing

of

G

Then

-LRB-

-RRB-

CA

is

the

set

of

cyclic

elements

of

the

matroid

MISn

-LRB-

G

-RRB-

and

-LRB-

-RRB-

V

-LRB-

G

-RRB-

n

CA

HA

-LSB-

DA

is

the

set

of

coloops

of

the

matroid

MISn

-LRB-

G

-RRB-

Proof

-LRB-

uses

-LRB-

-RRB-

and

-LRB-

-RRB-

-RRB-

Let

M

MISn

-LRB-

G

-RRB-

Let

us

prove

-LRB-

-RRB-

Since

A

is

a

V

-LCB-

maximal

induced

n

-LCB-

packing

of

G

clearly

B

V

-LSB-

A

-RSB-

is

a

base

of

the

matroid

M.

By

-LRB-

-RRB-

C

-LRB-

M

-RRB-

SfC

-LRB-

B

z

-RRB-

z

W

-LSB-

A

-RSB-

g.

By

C

-LRB-

B

z

-RRB-

CAz

is

the

set

of

the

last

vertices

of

passive

A

-LCB-

paths

starting

from

the

vertex

z.

By

CA

is

the

set

of

the

last

vertices

of

all

passive

A

-LCB-

paths

Therefore

CA

SfCAz

z

W

-LSB-

A

-RSB-

g

C

-LRB-

M

-RRB-

Now

-LRB-

-RRB-

follows

from

-LRB-

-RRB-

and

-LRB-

-RRB-

Induced

n

-LCB-

packing

structure

of

a

graph

Now

we

are

ready

to

prove

the

Structure

Theorem

for

induced

n

-LCB-

packings

in

a

graph

similar

to

the

Gallai

-LCB-

Edmonds

Structure

Theorem

for

matchings

Let

A

be

an

induced

n

-LCB-

packing

of

G

We

recall

some

notations

BA

is

the

set

of

vertices

of

G

that

can

be

reached

from

W

-LSB-

A

-RSB-

by

a

passive

A

-LCB-

path

BA

is

the

subgraph

of

G

induced

by

BA

-LRB-

i.e.

BA

G

-LSB-

BA

-RSB-

-RRB-

DA

V

-LRB-

G

-RRB-

n

BA

DA

G

n

B

A

SA

is

the

set

of

big

stars

of

A

in

BA

TA

SfT

-LRB-

S

-RRB-

S

SAg

-LSB-

W

-LSB-

A

-RSB-

HA

be

the

set

of

heads

of

big

stars

in

BA

-LRB-

i.e.

HA

fh

-LRB-

S

-RRB-

S

SAg

-RRB-

and

CA

BA

n

HA

and

C

A

BA

n

HA

We

also

recall

that

Cn

-LRB-

G

-RRB-

is

the

set

of

all

vertices

of

G

that

are

not

covered

by

at

least

one

V

-LCB-

maximum

induced

n

-LCB-

packing

of

G

Hn

-LRB-

G

-RRB-

is

the

set

of

all

vertices

in

V

-LRB-

G

-RRB-

nCn

-LRB-

G

-RRB-

adjacent

to

at

least

one

vertex

in

Cn

-LRB-

G

-RRB-

and

Dn

-LRB-

G

-RRB-

V

-LRB-

G

-RRB-

n

-LRB-

Hn

-LRB-

G

-RRB-

-LSB-

Cn

-LRB-

G

-RRB-

-RRB-

As

above

Cn

-LRB-

G

-RRB-

and

Dn

-LRB-

G

-RRB-

are

subgraphs

of

G

induced

by

the

vertex

sets

Cn

-LRB-

G

-RRB-

and

Dn

-LRB-

G

-RRB-

respectively

Let

G

be

an

arbitrary

graph

and

n

be

an

integer

at

least

Let

A

be

a

V

-LCB-

maximum

-LRB-

or

V

-LCB-

maximal

-RRB-

induced

n

-LCB-

packing

of

G

Then

-LRB-

see

Fig.

-RRB-

-LRB-

-RRB-

the

components

of

the

subgraph

Cn

-LRB-

G

-RRB-

of

G

are

ISn

-LCB-

critical

-LRB-

are

odd

clique

trees

-RRB-

-LRB-

-RRB-

the

subgraph

Dn

-LRB-

G

-RRB-

has

a

perfect

n

-LCB-

packing

-LRB-

-RRB-

CA

Cn

-LRB-

G

-RRB-

HA

Hn

-LRB-

G

-RRB-

DA

Dn

-LRB-

G

-RRB-

-LRB-

-RRB-

A

contains

-LRB-

-RRB-

a

near

perfect

matching

of

each

component

of

the

subgraph

Cn

-LRB-

G

-RRB-

-LRB-

-RRB-

a

perfect

n

-LCB-

packing

of

the

subgraph

Dn

-LRB-

G

-RRB-

and

-LRB-

-RRB-

a

set

of

jHn

-LRB-

G

-RRB-

j

disjoint

big

stars

such

that

their

heads

are

in

Hn

-LRB-

G

-RRB-

their

tails

are

in

Cn

-LRB-

G

-RRB-

and

each

component

of

Cn

-LRB-

G

-RRB-

contains

at

most

one

tail

of

all

these

big

stars

-LRB-

-RRB-

jW

-LSB-

A

-RSB-

j

minfjW

-LSB-

B

-RSB-

j

B

PISn

-LRB-

G

-RRB-

g

jOct

-LRB-

G

n

Hn

-LRB-

G

-RRB-

-RRB-

j

njHn

-LRB-

G

-RRB-

j.

Proof

-LRB-

uses

-LRB-

-RRB-

and

-LRB-

-RRB-

-RRB-

By

-LRB-

-RRB-

CA

is

the

set

of

cyclic

elements

of

the

matroid

MISn

-LRB-

G

-RRB-

By

-LRB-

-RRB-

CA

is

the

set

of

vertices

of

G

that

do

not

belong

to

at

least

one

base

of

MISn

-LRB-

G

-RRB-

Since

a

base

of

MISn

-LRB-

G

-RRB-

is

the

vertex

set

covered

by

a

V

-LCB-

maximal

induced

n

-LCB-

packing

we

have

CA

Cn

-LRB-

G

-RRB-

Therefore

by

HA

Hn

-LRB-

G

-RRB-

and

so

DA

Dn

-LRB-

G

-RRB-

Hence

-LRB-

-RRB-

holds

Now

-LRB-

-RRB-

follows

from

-LRB-

-RRB-

-LRB-

-RRB-

follows

from

-LRB-

-RRB-

follows

from

-LRB-

-RRB-

and

and

-LRB-

-RRB-

follows

from

Theorem

-LRB-

-RRB-

-LRB-

-RRB-

follows

from

and

-LRB-

-LRB-

-RRB-

The

following

conditions

are

equivalent

-LRB-

-RRB-

F

is

an

induced

ISn

-LCB-

critical

graph

n

-LRB-

-RRB-

F

is

an

induced

IS

-LCB-

critical

graph

and

-LRB-

-RRB-

F

is

an

odd

clique

tree

Proof

-LRB-

uses

and

-RRB-

By

an

odd

clique

tree

is

an

induced

ISn

-LCB-

critical

graph

n

and

an

induced

IS

-LCB-

critical

graph

i.e.

-LRB-

-RRB-

-RRB-

-LRB-

-RRB-

and

-LRB-

-RRB-

-RRB-

-LRB-

-RRB-

Clearly

-LRB-

-RRB-

-RRB-

-LRB-

-RRB-

Let

us

prove

-LRB-

-RRB-

-RRB-

-LRB-

-RRB-

Suppose

that

V

-LRB-

F

-RRB-

n

Cn

-LRB-

F

-RRB-

Then

by

F

n

x

does

not

have

a

perfect

induced

n

-LCB-

packing

and

so

F

is

not

an

induced

ISn

-LCB-

critical

graph

a

contradiction

Therefore

V

-LRB-

F

-RRB-

Cn

-LRB-

F

-RRB-

By

every

component

of

F

Cn

-LRB-

F

-RRB-

is

an

odd

clique

tree

Since

every

induced

ISn

-LCB-

critical

graph

is

connected

it

follows

that

F

is

an

odd

clique

tree

Theorem

follows

directly

from

and

Now

we

can

investigate

the

set

of

IS

-LCB-

obstacles

of

G

-LRB-

see

Theorem

-RRB-

Let

F

be

a

set

of

graphs

Given

a

graph

G

and

X

V

-LRB-

G

-RRB-

let

fG

-LRB-

X

-RRB-

denote

the

number

of

components

of

G

n

X

that

are

members

of

F

It

is

easy

to

see

that

fG

-LRB-

X

-RRB-

is

a

supermodular

function

of

X

i.e.

fG

-LRB-

X

-RRB-

fG

-LRB-

Y

-RRB-

fG

-LRB-

X

-LSB-

Y

-RRB-

fG

-LRB-

X

Y

-RRB-

for

every

two

vertex

subsets

X

and

Y

of

G.

Put

dG

-LRB-

X

-RRB-

Oct

-LRB-

G

n

X

-RRB-

njXj

and

m

minfjW

-LSB-

A

-RSB-

j

A

PISn

-LRB-

G

-RRB-

g

We

recall

that

X

V

-LRB-

G

-RRB-

is

an

IS

-LCB-

obstacle

if

dG

-LRB-

X

-RRB-

m.

Let

X

and

Y

be

IS

-LCB-

obstacles

in

G

Then

X

-LSB-

Y

and

X

Y

are

also

IS

-LCB-

obstacles

in

G.

Proof

-LRB-

uses

and

-RRB-

By

Oct

-LRB-

G

n

X

-RRB-

is

a

supermodular

function

of

X

Since

jXj

jY

j

jX

-LSB-

Y

j

jX

Y

j

the

function

dG

-LRB-

X

-RRB-

is

also

supermodular

i.e.

dG

-LRB-

X

-RRB-

dG

-LRB-

Y

-RRB-

dG

-LRB-

X

-LSB-

Y

-RRB-

dG

-LRB-

X

Y

-RRB-

Since

X

and

Y

are

IS

-LCB-

obstacles

in

G

we

have

dG

-LRB-

X

-RRB-

dG

-LRB-

Y

-RRB-

m.

By

m

maxfdG

-LRB-

Z

-RRB-

Z

V

-LRB-

G

-RRB-

g.

Therefore

dG

-LRB-

X

-LSB-

Y

-RRB-

m

and

dG

-LRB-

X

Y

-RRB-

m.

Hence

dG

-LRB-

X

-RRB-

dG

-LRB-

Y

-RRB-

dG

-LRB-

X

-LSB-

Y

-RRB-

dG

-LRB-

X

Y

-RRB-

From

the

above

inequalities

we

have

dG

-LRB-

X

-LSB-

Y

-RRB-

m

and

dG

-LRB-

X

Y

-RRB-

m

and

so

both

X

-LSB-

Y

and

X

Y

are

IS

-LCB-

obstacles

in

G.

Let

Zn

-LRB-

G

-RRB-

denote

the

set

of

all

ISn

-LCB-

obstacles

in

G.

Let

Ln

-LRB-

G

-RRB-

-LRB-

Zn

-LRB-

G

-RRB-

-RRB-

denote

the

set

Zn

-LRB-

G

-RRB-

partially

ordered

by

the

inclusion

operation

Then

-LRB-

-RRB-

Ln

-LRB-

G

-RRB-

is

a

sublattice

of

of

the

lattice

of

all

subsets

of

V

-LRB-

G

-RRB-

under

inclusion

-LRB-

-RRB-

Hn

-LRB-

G

-RRB-

is

the

minimum

element

of

Ln

-LRB-

G

-RRB-

and

-LRB-

-RRB-

if

X

Ln

-LRB-

G

-RRB-

then

X

n

Hn

-LRB-

G

-RRB-

Ln

-LRB-

Dn

-LRB-

G

-RRB-

-RRB-

Proof

-LRB-

uses

and

-RRB-

Clearly

-LRB-

-RRB-

follows

from

Let

us

prove

-LRB-

-RRB-

Put

H

Hn

-LRB-

G

-RRB-

and

W

W

-LSB-

A

-RSB-

Let

A

be

a

V

-LCB-

maximum

induced

n

-LCB-

packing

in

G.

By

H

HA

and

Cn

-LRB-

G

-RRB-

C

A.

By

Cmp

-LRB-

C

A

-RRB-

fNa

a

TAg

Let

Z

TA

and

Y

H.

Put

N

-LRB-

Z

-RRB-

fNa

a

Zg

N

h

fNa

a

T

-LRB-

Shg

for

h

H

and

N

Y

-LSB-

fN

h

h

Y

g.

Obviously

jN

-LRB-

Z

-RRB-

j

jZj

Since

h

H

clearly

jT

-LRB-

Shj

n

and

so

jN

hj

jT

-LRB-

Shj

n.

Therefore

jN

Y

j

njY

j.

Clearly

Oct

-LRB-

G

n

Y

-RRB-

N

-LRB-

W

-RRB-

-LSB-

N

Y

and

so

jOct

-LRB-

G

n

Y

-RRB-

j

jN

-LRB-

W

-RRB-

j

jN

Y

j

jW

j

njY

j.

Let

X

ae

H

It

is

sufficient

to

prove

that

X

is

not

an

ISn

-LCB-

obstacle

in

G

i.e.

that

jOct

-LRB-

G

n

X

-RRB-

j

njXj

jW

j.

Suppose

on

the

contrary

that

jOct

-LRB-

G

n

X

-RRB-

njXj

jW

j

Since

jOct

-LRB-

G

n

X

-RRB-

j

njXj

jW

j

we

have

jOct

-LRB-

G

n

X

-RRB-

j

njXj

jW

j

Then

Oct

-LRB-

G

n

X

-RRB-

N

-LRB-

W

-RRB-

-LSB-

NX

Put

OX

-LSB-

fK

K

Oct

-LRB-

G

n

X

-RRB-

g.

Let

h

H

n

X

Then

h

OX

and

h

is

adjacent

to

no

vertex

in

OX

Since

h

is

a

vertex

of

BA

there

exists

a

passive

A

-LCB-

path

wP

z

containing

h

so

that

h

P

and

w

W

By

the

above

equation

w

W

OX

Therefore

wPh

has

an

edge

connecting

h

with

a

vertex

v

in

OX

-LSB-

X

Since

h

adjacent

to

no

vertex

in

OX

clearly

v

X

Since

-LRB-

h

v

-RRB-

E

-LRB-

G

-RRB-

n

A

the

subpath

wPv

of

P

is

a

passive

A

-LCB-

path

and

so

the

last

edge

-LRB-

v

-RRB-

of

wPv

is

an

edge

of

the

big

star

Sv

with

the

head

v

and

a

tail

Therefore

the

tail

preceeds

the

head

v

of

Sv

in

wPv

Hence

wPv

is

not

a

passive

A

-LCB-

path

a

contradiction

Now

-LRB-

-RRB-

follows

from

-LRB-

-RRB-

and

-LRB-

-RRB-

Polynomial

algorithm

for

packing

induced

stars

in

a

graph

Let

A

be

an

induced

n

-LCB-

packing

of

G.

Let

F

be

a

subgraph

of

G

We

use

the

following

notation

similar

to

that

in

Section

SA

-LRB-

F

-RRB-

is

the

set

of

big

stars

of

A

in

F

TA

-LRB-

F

-RRB-

-LSB-

fT

-LRB-

S

-RRB-

S

SA

-LRB-

F

-RRB-

g

-LSB-

-LRB-

W

-LSB-

A

-RSB-

F

-RRB-

HA

-LRB-

F

-RRB-

fh

-LRB-

S

-RRB-

S

SA

-LRB-

F

-RRB-

g

and

FA

F

n

HA

-LRB-

F

-RRB-

A

subgraph

F

of

G

is

called

an

A

-LCB-

subgraph

if

-LRB-

-RRB-

W

-LSB-

A

-RSB-

V

-LSB-

F

-RSB-

-LRB-

-RRB-

if

S

is

a

star

of

A

in

F

then

S

is

either

a

big

star

or

a

small

star

-LRB-

-RRB-

if

S

is

a

star

of

A

and

S

F

then

S

F

-LRB-

-RRB-

every

component

of

F

contains

exactly

one

vertex

from

W

-LSB-

A

-RSB-

-LRB-

-RRB-

every

component

of

FA

is

an

odd

clique

tree

-LRB-

-RRB-

every

component

of

FA

contains

exactly

one

vertex

from

TA

-LRB-

F

-RRB-

-LRB-

-RRB-

if

K

is

a

component

of

FA

then

K

n

TA

-LRB-

F

-RRB-

has

a

perfect

matching

M

-LRB-

K

-RRB-

consisting

of

small

stars

of

A

-LRB-

i.e.

M

-LRB-

K

-RRB-

-RRB-

Let

us

now

give

an

-LRB-

informal

-RRB-

description

of

an

algorithm

that

for

a

given

graph

G

finds

a

vertex

maximum

induced

n

-LCB-

packing

A

as

well

as

Hn

-LRB-

G

-RRB-

A

similar

algorithm

can

be

used

to

find

for

X

V

-LRB-

G

-RRB-

an

induced

n

-LCB-

star

packing

that

covers

X

-LRB-

if

any

-RRB-

The

general

step

deals

with

the

current

information

-LRB-

A

F

H

;P

-RRB-

where

A

is

an

induced

n

-LCB-

packing

A

F

is

an

A

-LCB-

subgraph

of

G

H

HA

-LRB-

F

-RRB-

and

P

fPx

x

V

-LRB-

FA

-RRB-

g

where

Px

is

a

passive

A

-LCB-

path

in

F

terminating

at

x

In

the

first

step

A

-LRB-

and

so

W

-LSB-

A

-RSB-

V

-LRB-

G

-RRB-

-RRB-

and

F

is

the

subgraph

of

G

with

V

-LRB-

F

-RRB-

W

-LSB-

A

-RSB-

V

-LRB-

G

-RRB-

and

E

-LRB-

F

-RRB-

and

so

H

HA

-LRB-

F

-RRB-

and

Px

x

for

x

V

-LRB-

G

-RRB-

In

every

step

we

have

one

of

the

following

goals

-LRB-

-RRB-

enlarge

F

i.e.

find

A

-LCB-

subgraph

F

of

G

such

that

F

ae

F

and

-LRB-

-RRB-

enlarge

A

by

finding

an

active

A

-LCB-

trail

Q

and

putting

so

that

V

-LRB-

A

-RRB-

ae

V

-LRB-

-RRB-

If

we

can

not

reach

either

of

these

goals

then

this

step

is

final

the

current

induced

n

-LCB-

packing

A

of

this

step

is

V

-LCB-

maximum

Now

we

shall

describe

a

general

step

of

the

algorithm

General

step

S.

Find

-LRB-

if

any

-RRB-

an

edge

e

-LRB-

x

y

-RRB-

of

G

having

the

following

property

-LRB-

p

-RRB-

x

FA

and

-LRB-

x

y

-RRB-

-LRB-

G

-RRB-

n

E

-LRB-

F

-RRB-

If

there

is

no

such

edge

then

stop

A

is

V

-LCB-

maximum

and

H

HA

-LRB-

F

-RRB-

Hn

-LRB-

G

-RRB-

So

suppose

we

have

found

an

edge

e

-LRB-

x

y

-RRB-

with

the

property

-LRB-

p

-RRB-

Let

Px

wPx

There

are

three

options

for

y

-LRB-

-RRB-

y

HA

-LRB-

F

-RRB-

-LRB-

-RRB-

y

V

-LRB-

F

-RRB-

n

HA

-LRB-

F

-RRB-

and

-LRB-

-RRB-

y

V

-LRB-

G

-RRB-

n

V

-LRB-

F

-RRB-

In

case

-LRB-

-RRB-

put

F

F

-LSB-

e

H

H

and

P

P.

Let

us

consider

case

-LRB-

-RRB-

Suppose

that

x

and

y

belong

to

different

components

of

FA

Then

find

an

active

A

-LCB-

trail

Q.

-LSB-

By

an

active

A

-LCB-

trail

exists

The

proof

of

shows

how

to

find

it

by

using

the

passive

A

-LCB-

path

wPx

-RSB-

Now

suppsoe

that

x

and

y

belong

to

the

same

component

K

of

FA

Since

F

is

an

A

-LCB-

subgraph

of

G

the

component

K

is

an

odd

clique

tree

Hence

we

can

find

a

-LRB-

unique

-RRB-

chain

L

-LRB-

x

-RRB-

Bs

-LRB-

bs

y

-RRB-

where

Bi

is

a

block

of

K

and

bi

bi

V

-LRB-

Bi

-RRB-

i

s

If

G

-LSB-

L

-RSB-

is

a

complete

graph

put

F

F

-LSB-

G

-LSB-

L

-RSB-

H

H

and

P

P

If

G

-LSB-

L

-RSB-

is

not

a

complete

graph

then

find

an

active

A

-LCB-

trail

Q.

-LSB-

By

an

active

A

-LCB-

trail

exists

and

the

proof

of

shows

how

to

find

it

by

using

the

passive

A

-LCB-

path

wPx

-RSB-

In

case

-LRB-

-RRB-

find

the

star

Sy

containing

y

-LSB-

clearly

such

star

exists

and

F

Sy

Sy

can

be

either

a

small

star

or

an

intermediate

star

or

a

big

star

If

Sy

is

a

non-small

star

then

y

can

be

either

a

tail

or

the

head

of

Sy

Suppose

that

Sy

is

a

small

star

of

A

say

G

-LSB-

y

z

-RSB-

Then

check

whether

-LRB-

x

z

-RRB-

is

an

edge

of

G

If

-LRB-

x

z

-RRB-

E

-LRB-

G

-RRB-

then

put

F

F

-LSB-

G

-LSB-

x

y

z

-RSB-

H

H

and

P

P

-LSB-

fPy

Pzg

where

Py

wP

x

z

y

and

Py

wP

x

y

z

If

-LRB-

x

z

-RRB-

E

-LRB-

G

-RRB-

then

Q

wP

x

y

is

a

active

A

-LCB-

path

Now

suppose

that

Sy

is

a

non-small

star

Let

h

h

-LRB-

Sy

-RRB-

Suppose

that

y

is

a

tail

of

Sy

Then

Q

wP

x

y

h

is

a

-LCB-

active

A

-LCB-

path

Suppose

that

y

is

the

head

of

Sy

y

h

Then

find

a

tail

t

of

Sy

adjacent

to

x.

Suppose

that

such

tail

exists

Then

Q

wP

x

t

h

is

a

-LCB-

active

A

-LCB-

path

Now

suppose

that

no

tail

is

adjacent

to

x

If

Sy

is

an

intermediate

star

then

Q

wPxey

is

an

-LCB-

active

A

-LCB-

path

If

Sy

is

a

big

star

then

put

F

F

-LSB-

Sy

-LSB-

e

H

H

-LSB-

y

and

P

P

-LSB-

fPt

t

T

-LRB-

Sy

-RRB-

where

Pt

wP

x

y

t.

Thus

we

have

one

of

the

following

two

outcomes

-LRB-

-RRB-

a

new

A

-LCB-

subgraph

F

such

that

F

ae

F

and

-LRB-

-RRB-

an

active

A

-LCB-

trail

Q

In

case

-LRB-

-RRB-

put

A

In

case

-LRB-

-RRB-

let

-LRB-

Q

-RRB-

and

F

be

the

subgraph

with

V

-LRB-

F

-RRB-

W

-LSB-

-RSB-

E

-LRB-

F

-RRB-

H

and

P

Px

x

V

-LRB-

F

-RRB-

where

Px

x

Now

repeat

step

S

with

the

new

information

-LRB-

A

F

H

;P

-RRB-

-LRB-

F

H

P

-RRB-

Clearly

the

algorithm

described

above

is

polynomial

although

far

from

being

the

best

For

example

in

case

-LRB-

-RRB-

a

better

F

can

be

found

by

an

appropriate

modification

of

F

Also

it

is

sometimes

easier

to

find

an

augmenting

A

-LCB-

trail

instead

of

active

A

-LCB-

quasi

-LCB-

path

This

algorithm

can

be

used

to

give

alternative

proofs

of

the

main

results

A

more

efficient

algorithm

and

an

alternative

proofs

of

the

main

results

based

on

this

algorithm

will

be

given

in

another

paper

Acknowledgment

I

am

very

thankful

to

a

referee

for

a

very

careful

reading

of

the

paper

and

very

useful

remarks

References

-LSB-

-RSB-

A.

Amahashi

and

M.

Kano

Factors

with

given

components

Discrete

Math.

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

J.A.

Bondy

and

U.S.R.

Murty

Graph

theory

with

applications

North

Holland

Amsterdam

-LSB-

-RSB-

G.

Cornu?ejols

and

D.

Hartvigsen

An

extension

of

matching

theory

J.

Combin

Theory

B

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

P.

Hell

and

D.

Kirkpatrick

On

the

complexity

of

general

graph

factor

problems

SIAM

J.

Comput

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

P.

Hell

and

D.

Kirkpatrick

Packing

by

complete

bipartite

graphs

SIAM

J.

Algebraic

Discrete

Math

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

P.

Hell

and

D.

Kirkpatrick

Star

factors

and

star

packings

TR

Department

of

Computer

Science

Simon

Fraser

University

-LSB-

-RSB-

A.K.

Kelmans

Matroids

and

combinatorial

optimization

problems

Institute

of

Control

Sciences

Moscow

-LSB-

-RSB-

A.K.

Kelmans

M.V.

Lomonosov

V.P.

Polessky

On

the

minimal

covering

in

a

matroid

Problemi

peredachi

informacii

-LRB-

-RRB-

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

A.K.

Kelmans

and

V.P.

Polessky

Extremal

sets

and

covering

and

packing

problems

in

a

matroid

In

Issled

po

prikladnoy

teorii

grafov

Novosibirsk

-LCB-

-LRB-

see

also

Amer

Math

Soc

Transl

-LRB-

-RRB-

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

M.

Las

Vergnas

An

extension

of

Tutte

's

-LCB-

factor

theorem

Discrete

Math

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

M.

Loebl

and

S.

Poljak

On

matroids

induced

by

packing

subgraphs

J.

Combin

Theory

B

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

M.

Loebl

and

S.

Poljak

Subgraph

packing

-LCB-

A

survey

In

Topics

in

Combinatorucs

and

Graph

Theory

Essays

in

Honour

of

Gerhard

Ringel

Phisica

Verlag

Wursburg

-LCB-

-LSB-

-RSB-

M.

Loebl

and

S.

Poljak

Efficient

subgraph

packing

J.

Combin

Theory

B

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

L.

Lovasz

and

M.

Plummer

Matching

theory

North-Holland

Amsterdam

-LSB-

-RSB-

D.J.

Welsh

Matroid

Theory

Academic

Press

New

York

factor graph matching matroid optimization packing stars