aarjaneiro/planetmath huggingface dataset:
edge covering

edge covering

Let

G

be a graph. An

edge covering

C

on

G

is a subset of the vertices of

G

such that each edge in

G

is

incident

with at least one vertex in

C

.

For any graph, the

vertex set

is a trivial edge covering. Generally, we are more interested in

minimal coverings

. A minimal edge covering is simply an edge covering of the least possible .

Title

edge covering

Canonical

name

EdgeCovering

Date of creation

2013-03-22 12:40:02

Last modified on

2013-03-22 12:40:02

Owner

vampyr (22)

Last modified by

vampyr (22)

Numerical id

8

Author

vampyr (22)

Entry type

Definition

Classification

msc 05C70

Related topic

Matching

Defines

minimal edge covering

Generated on Fri Feb 9 07:55:39 2018 by

LaTeXML
matching

matching

Let

G

be a graph. A

matching

M

on

G

is a subset of the edges of

G

such that each vertex in

G

is

incident

with no more than one edge in

M

.

It is easy to find a matching on a graph; for example, the empty set will always be a matching. Typically, the most interesting matchings are

maximal matchings

. A maximal matching on a graph

G

is simply a matching of the largest possible size.

A

perfect matching

is a matching that

saturates

every vertex.

Title

matching

Canonical name

Matching

Date of creation

2013-03-22 12:40:00

Last modified on

2013-03-22 12:40:00

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 05C70

Related topic

MaximalMatchingminimalEdgeCoveringTheorem

Related topic

Matching

Related topic

EdgeCovering

Related topic

Saturate

Defines

maximal matching

Defines

perfect matching

Generated on Fri Feb 9 07:55:41 2018 by

LaTeXML
maximal matching/minimal edge covering theorem

maximal matching/minimal edge covering theorem

Theorem

Let

G

be a graph. If

M

is a

matching

on

G

, and

C

is an

edge covering

for

G

, then

|

M

|

≤

|

C

|

.

Proof

Consider an arbitrary matching

M

on

G

and an arbitrary edge covering

C

on

G

. We will attempt to construct a one-to-one function

f

:

M

→

C

.

Consider some edge

e

∈

M

. At least one of the vertices that

e

joins must be in

C

, because

C

is an edge covering and hence every edge is

incident

with some vertex in

C

. Call this vertex

v

e

, and let

f

⁢

(

e

)

=

v

e

.

Now we will show that

f

one-to-one. Suppose we have two edges

e

1

,

e

2

∈

M

where

f

⁢

(

e

1

)

=

f

⁢

(

e

2

)

=

v

. By the

definition

of

f

,

e

1

and

e

2

must both be incident with

v

. Since

M

is a matching, however, no more than one edge in

M

can be incident with any given vertex in

G

. Therefore

e

1

=

e

2

, so

f

is one-to-one.

Hence we now have that

|

M

|

≤

|

C

|

.

Corollary

Let

G

be a graph. Let

M

and

C

be a matching and an edge covering on

G

, respectively. If

|

M

|

=

|

C

|

, then

M

is a maximal matching and

C

is a minimal edge covering.

Proof

Suppose

M

is not a maximal matching. Then, by definition, there exists another matching

M

′

where

|

M

|

<

|

M

′

|

. But then

|

M

′

|

>

|

C

|

, which violates the above

theorem

.

Likewise, suppose

C

is not a minimal edge covering. Then, by definition, there exists another covering

C

′

where

|

C

′

|

<

|

C

|

. But then

|

C

′

|

<

|

M

|

, which violates the above theorem.

Title

maximal matching/minimal edge covering theorem

Canonical

name

MaximalMatchingminimalEdgeCoveringTheorem

Date of creation

2013-03-22 12:40:05

Last modified on

2013-03-22 12:40:05

Owner

mps (409)

Last modified by

mps (409)

Numerical id

7

Author

mps (409)

Entry type

Theorem

Classification

msc 05C70

Related topic

MaximumFlowminimumCutTheorem

Related topic

Matching

Generated on Fri Feb 9 07:55:45 2018 by

LaTeXML
Petersen theorem

Petersen theorem

Every finite , 3-regular (

http://planetmath.org/

Valency

), 2-edge

connected graph

has a

complete matching

.

Proof.

Using the notations from the

Tutte theorem

, we have to prove that for all

X

⊆

V

⁢

(

G

)

the

inequality

c

p

⁢

(

G

-

X

)

≤

|

X

|

holds. There are at least

3

edges running between

X

and an odd component of

G

-

X

: there cannot be one edge, since

G

is 2-edge connected, and there also cannot be two edges, because three edges start from all vertices of an odd component, so the number of edges leaving an odd component is odd. Let

t

be the number of all edges between

X

and the odd components of

G

-

X

. Now we have

t

≥

3

⁢

c

p

⁢

(

G

-

X

)

. But

G

is 3-regular, thus

t

≤

3

⁢

|

X

|

. This gives

c

p

⁢

(

G

-

X

)

≤

|

X

|

.
∎

Title

Petersen theorem

Canonical name

PetersenTheorem

Date of creation

2013-03-22 14:06:07

Last modified on

2013-03-22 14:06:07

Owner

scineram (4030)

Last modified by

scineram (4030)

Numerical id

15

Author

scineram (4030)

Entry type

Theorem

Classification

msc 05C70

Generated on Fri Feb 9 07:55:47 2018 by

LaTeXML
alternate proof of Mantel’s theorem

alternate proof of Mantel’s theorem

Let

G

be a triangle-free graph of order

n

. For each edge

x

⁢

y

of

G

we consider the

neighbourhoods

(

http://planetmath.org/NeighborhoodOfAVertex

)

Γ

⁢

(

x

)

and

Γ

⁢

(

y

)

of

G

.
Since

G

is triangle-free, these are disjoint.

This is only possible if the sum of the degrees of

x

and

y

is less
than or equal to

n

. So for each edge

x

⁢

y

we get the

inequality

δ

⁢

(

x

)

+

δ

⁢

(

y

)

≤

n

Summing these inequalities for all edges of

G

gives us

Σ

x

∈

V

⁢

(

G

)

⁢

(

δ

⁢

(

x

)

)

2

≤

n

⁢

|

E

⁢

(

G

)

|

(The

left hand side

is a sum of

δ

⁢

(

x

)

where each edge incident with

x

contributes one term and their are

δ

⁢

(

x

)

such edges.)

Since

Σ

x

∈

V

⁢

(

G

)

⁢

δ

⁢

(

x

)

=

2

⁢

|

E

⁢

(

G

)

|

, we get

4

⁢

|

E

⁢

(

G

)

|

2

=

(

Σ

x

∈

V

⁢

(

G

)

⁢

δ

⁢

(

x

)

)

2

and applying the

Cauchy-Schwarz inequality

gives

4

⁢

|

E

⁢

(

G

)

|

2

≤

n

⁢

Σ

x

∈

V

⁢

(

G

)

⁢

(

δ

⁢

(

x

)

)

2

≤

n

2

⁢

|

E

⁢

(

G

)

|

.

So we conclude that for a triangle-free graph

G

,

|

E

⁢

(

G

)

|

≤

n

2

4

Title

alternate proof of Mantel’s theorem

Canonical name

AlternateProofOfMantelsTheorem

Date of creation

2013-03-22 17:56:35

Last modified on

2013-03-22 17:56:35

Owner

lieven (1075)

Last modified by

lieven (1075)

Numerical id

5

Author

lieven (1075)

Entry type

Proof

Classification

msc 05C75

Classification

msc 05C69

Generated on Fri Feb 9 07:55:52 2018 by

LaTeXML
alternative definition of a multigraph

alternative definition of a multigraph

Many authors tried to formalize the notation of a graph. This problem is relatively simple if we allow at most

1

edge between vertices. But for

multigraphs

, i.e. graphs with many edges (possibly infinitely many) between vertices this tends to be problematic formally. We wish to give an alternative definition, which uses so called

symmetric power

(

http://planetmath.org/SymmetricPower

).

Definition.

A

multigraph

or

non-oriented graph

is a triple

G

=

(

V

,

E

,

τ

)

where

V

is a

nonempty set

whose

elements

are called vertices,

E

is a set whose elements are called edges and

τ

:

E

→

V

s

⁢

y

⁢

m

2

is a

function

which takes every edge to a pair of vertices called ends of this edge. On the right side we have a symmetric power (

http://planetmath.org/SymmetricPower

) of

V

to ensure that the order of ends is not important.

This definition allows loops and even

infinite

number of edges between two vertices and is one of the most general and formal.

Title

alternative definition of a multigraph

Canonical name

AlternativeDefinitionOfAMultigraph

Date of creation

2013-03-22 19:16:54

Last modified on

2013-03-22 19:16:54

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Definition

Classification

msc 05C75

Generated on Fri Feb 9 07:55:53 2018 by

LaTeXML
line graph

line graph

Let

G

be a graph. The

line graph

of

G

is denoted by

L

⁢

(

G

)

and is defined as follows. The vertices of

L

⁢

(

G

)

are the edges of

G

. Two vertices of

L

⁢

(

G

)

are

connected

by an edge if and only if the corresponding edges in

G

form a path (which must be a directed path if

G

is a

digraph

).

Example 1

.

Let

G

be the

undirected graph

on the

vertex set

V

⁢

(

G

)

=

{

a

,

b

,

c

,

d

}

and

edge set

E

⁢

(

G

)

=

{

a

⁢

b

,

b

⁢

c

,

c

⁢

d

,

b

⁢

d

}

. Then the line graph

L

⁢

(

G

)

has vertex set

V

⁢

(

L

⁢

(

G

)

)

=

{

a

⁢

b

,

b

⁢

c

,

c

⁢

d

,

b

⁢

d

}

. Since

a

⁢

b

and

b

⁢

c

are

incident

in

G

, they are connected by an edge in

L

⁢

(

G

)

. We label this edge by the unique walk from

a

to

c

determined by these edges, namely

a

⁢

b

⁢

c

. With this notation, the edge set is

E

⁢

(

L

⁢

(

G

)

)

=

{

a

⁢

b

⁢

c

,

a

⁢

b

⁢

d

,

b

⁢

c

⁢

d

,

b

⁢

d

⁢

c

,

c

⁢

b

⁢

d

}

.

\xymatrix

⁢

&

⁢

a

⁢

\ar

⁢

@

-

[

d

]

a

⁢

b

⁢

&

⁢

&

⁢

&

⁢

a

⁢

b

⁢

\ar

⁢

@

-

[

d

⁢

l

]

a

⁢

b

⁢

c

⁢

\ar

⁢

@

-

[

d

⁢

r

]

a

⁢

b

⁢

d

⁢

&

⁢

&

⁢

b

⁢

\ar

⁢

@

-

[

d

⁢

l

]

b

⁢

c

⁢

\ar

⁢

@

-

[

d

⁢

r

]

b

⁢

d

⁢

&

⁢

&

⁢

b

⁢

c

⁢

\ar

⁢

@

-

[

r

⁢

r

]

b

⁢

c

⁢

d

⁢

\ar

⁢

@

-

[

d

⁢

r

]

c

⁢

b

⁢

d

⁢

&

⁢

&

⁢

b

⁢

d

⁢

\ar

⁢

@

-

[

d

⁢

l

]

b

⁢

d

⁢

c

⁢

c

⁢

\ar

⁢

@

-

[

r

⁢

r

]

c

⁢

d

⁢

&

⁢

&

⁢

d

⁢

&

⁢

&

⁢

c

⁢

d

Above we display the graphs

G

(left) and

L

⁢

(

G

)

(right).

Part of the reason for

interest

in line graphs is the following observation:

Proposition

.

If a graph is Eulerian, then its line graph is

Hamiltonian

.

This follows immediately from the fact that a sequence of incident edges in

G

is a sequence of incident vertices in

L

⁢

(

G

)

.

Example 2

.

Let

B

n

be the

de Bruijn digraph

on binary strings of length

n

. Then for

n

≥

2

, the next de Bruijn digraph can be obtained by taking the line graph of the previous one, that is,

B

n

+

1

=

L

⁢

(

B

n

)

.

Title

line graph

Canonical

name

LineGraph

Date of creation

2013-03-22 16:24:15

Last modified on

2013-03-22 16:24:15

Owner

mps (409)

Last modified by

mps (409)

Numerical id

4

Author

mps (409)

Entry type

Definition

Classification

msc 05C75

Related topic

DeBruijnDigraph

Generated on Fri Feb 9 07:56:08 2018 by

LaTeXML
multigraph

multigraph

A

multigraph

is a graph in which we allow more than one edge to join a pair of vertices. Two or more edges that join a pair of vertices are called

parallel edges

. Every graph, then, is a multigraph, but not all multigraphs are graphs.
Some authors define the concept of a graph by excluding graphs with

multiple
edges

or loops. Then if they want to consider more general graphs the
multigraph is introduced. Usually, such graphs have no loops.
Formally, a multigraph

G

=

(

V

,

E

)

is a pair, where

E

=

(

V

(

2

)

,

f

)

is a multiset for which

f

⁢

(

x

,

x

)

=

0

and

V

(

2

)

is the set of unordered pairs
of

V

.

A multigraph can be used to a matrix whose entries are

nonnegative integers

. To do this, suppose that

A

=

(

a

i

⁢

j

)

is an

m

×

n

matrix of nonnegative integers.
Let

V

=

S

∪

T

, where

S

=

{

1

,

…

,

m

}

and

T

=

{

1

′

,

…

,

n

′

}

and connect vertex

i

∈

S

to vertex

j

′

∈

T

with

a

i

⁢

j

edges.

Title

multigraph

Canonical

name

Multigraph

Date of creation

2013-03-22 11:57:57

Last modified on

2013-03-22 11:57:57

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 05C75

Synonym

parallel edge

Related topic

Graph

Related topic

Subgraph

Related topic

GraphHomomorphism

Related topic

Pseudograph

Related topic

Quiver

Related topic

AxiomsOfMetacategoriesAndSupercategories

Generated on Fri Feb 9 07:56:19 2018 by

LaTeXML
pseudograph

pseudograph

A

pseudograph

is a graph that allows both

parallel edges

and loops.
Formally,

G

=

(

V

,

E

)

is a pseudograph , if

E

is a multiset

(

V

(

2

)

,

f

)

where

V

(

2

)

is the set of unordered pairs of

V

.

Title

pseudograph

Canonical

name

Pseudograph

Date of creation

2013-03-22 11:58:00

Last modified on

2013-03-22 11:58:00

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

9

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 05C75

Related topic

Graph

Related topic

Multigraph

Related topic

LoopOfAGraph

Related topic

Subgraph

Related topic

GraphHomomorphism

Generated on Fri Feb 9 07:56:24 2018 by

LaTeXML
labeled graph

labeled graph

Let

G

=

(

V

,

E

)

be a graph with

vertex set

V

and

edge set

E

. A

labeling

of

G

is a

partial function

ℓ

:

V

∪

E

→

L

for some set

L

. For every

x

in the domain of

ℓ

, the

element

ℓ

⁢

(

x

)

∈

L

is called the

label

of

x

. Three of the most common types of labelings of a graph

G

are

•

total labeling

:

ℓ

is a

total function

(defined for all of

V

∪

E

),

•

vertex labeling

: the domain of

ℓ

is

V

, and

•

edge labeling

: the domain of

ℓ

is

E

.

Usually,

L

above is assumed to be a set of integers. A

labeled graph

is a pair

(

G

,

ℓ

)

where

G

is a graph and

ℓ

is a labeling of

G

.

An example of a labeling of a graph is a

coloring

of a graph. Uses of graph labeling outside of

combinatorics

can be found in areas such as order theory,

language

theory, and

proof theory

. A

proof tree

, for

instance

, is really a

labeled tree

, where the labels of vertices are

formulas

, and the labels of edges are

rules of inference

.

Remarks

.

•

Every labeling of a graph can be extended to a total labeling.

•

The notion of labeling can be easily extended to

digraphs

,

multigraphs

, and

pseudographs

.

Title

labeled graph

Canonical name

LabeledGraph

Date of creation

2013-03-22 17:38:19

Last modified on

2013-03-22 17:38:19

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 05C78

Synonym

labelled graph

Synonym

graph labelling

Synonym

labelling

Synonym

vertex labelling

Synonym

edge labelling

Synonym

total labelling

Synonym

labelled tree

Synonym

labelled multigraph

Synonym

labelled pseudograph

Defines

graph labeling

Defines

labeling

Defines

vertex labeling

Defines

edge labeling

Defines

total labeling

Defines

labeled tree

Defines

labeled multigraph

Defines

labeled pseudograph

Generated on Fri Feb 9 07:56:35 2018 by

LaTeXML
example of a probabilistic proof

example of a probabilistic proof

Our example is the existence of

k

-

paradoxical tournaments

. The proof hinges upon the following basic probabilistic

inequality

, for any events

A

and

B

,

P

⁢

(

A

∪

B

)

≤

P

⁢

(

A

)

+

P

⁢

(

B

)

Theorem 1

.

For every

k

, there exists a

k

-paradoxical tournament.

Proof.

We will construct a

tournament

T

on

n

vertices. We will show that for

n

large enough, a

k

-paradoxical tournament must exist. The

probability space

in question is all possible directions of the arrows of

T

, where each arrow can point in either direction with

probability

1

/

2

, independently of any other arrow.

We say that a set

K

of

k

vertices is

arrowed

by a vertex

v

0

outside the set if every arrow between

v

0

to

w

i

∈

K

points from

v

0

to

w

i

, for

i

=

1

,

…

,

k

. Consider a fixed set

K

of

k

vertices and a fixed vertex

v

0

outside

K

. Thus, there are

k

arrows from

v

0

to

K

, and only one

arrangement

of these arrows permits

K

to be arrowed by

v

0

, thus

P

⁢

(

K

⁢

is arrowed by

⁢

v

0

)

=

1

2

k

.

The complementary event, is therefore,

P

⁢

(

K

⁢

is

not

arrowed by

⁢

v

0

)

=

1

-

1

2

k

.

By independence, and because there are

n

-

k

vertices outside of

K

,

P

⁢

(

K

⁢

is not arrowed by

any

vertex

)

=

(

1

-

1

2

k

)

n

-

k

.

(1)

Lastly, since there are

(

n

k

)

sets of cardinality

k

in

T

, we employ the inequality mentioned above to obtain that for the union of all events of the form in equation (

1

)

P

⁢

(

Some set of

k

vertices is not arrowed by any vertex

)

≤

(

n

k

)

⁢

(

1

-

1

2

k

)

n

-

k

.

If the probability of this last event is less than

1

for some

n

, then there must exist a

k

-paradoxical tournament of

n

vertices. Indeed there is such an

n

, since

(

n

k

)

⁢

(

1

-

1

2

k

)

n

-

k

=

1

k

!

⁢

n

⁢

(

n

-

1

)

⁢

⋯

⁢

(

n

-

k

+

1

)

⁢

(

1

-

1

2

k

)

n

-

k

<

1

k

!

⁢

n

k

⁢

(

1

-

1

2

k

)

n

-

k

Therefore, regarding

k

as fixed while

n

tends to

infinity

, the right-hand-side above tends to zero. In particular, for some

n

it is less than

1

, and the result follows.
∎

Title

example of a

probabilistic proof

Canonical

name

ExampleOfAProbabilisticProof

Date of creation

2013-03-22 13:06:18

Last modified on

2013-03-22 13:06:18

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

13

Author

bbukh (348)

Entry type

Example

Classification

msc 05C80

Generated on Fri Feb 9 07:56:38 2018 by

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probabilistic method

probabilistic method

The

probabilistic method

was pioneered by Erdős Pál (known to Westerners as Paul Erdős) and initially used for solving problems in

graph theory

, but has acquired ever wider applications. Broadly, the probabilistic method is somewhat the opposite of

extremal graph theory

. Instead of considering how a graph can behave in the most extreme case, we consider how a

collection

of graphs behave “on

average

”, whereby we can formulate a

probability space

. The fruits reaped by this method are often raw

existence theorems

, usually deduced from the fact that the nonexistence of whatever of graph would a zero

probability

. For

instance

, by means of the probabilistic method, Erdős proved the

existence

of a graph of arbitrarily high girth and

chromatic number

, a very counterintuitive result. Graphs tend to get enormous as the

chromatic number and girth

increase, thereby severely hindering necessary computations to explicitly construct them, so an existence theorem is most welcome.

In all honesty,

probabilistic proofs

are nothing more than counting proofs in disguise, since determining the probabilities of

interest

will invariably involve detailed counting

arguments

. In fact, we

could

remove from any probabilistic proof any mention of a probability space, although the result may be significanltly less transparent. Also, the advantage of using probability is that we can employ all the machinery of probability theory.

Markov chains

,

martingales

,

expectations

(

http://planetmath.org/ExpectedValue

), probabilistic inequalities, and many other results, all become the tools of the trade in dealing with seemingly static objects of

combinatorics

and

number theory

.

References

1

Noga Alon and Joel H. Spencer.

The probabilistic method

.

John Wiley & Sons, Inc., second edition, 2000.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0996.05001

Zbl
0996.05001.

2

Paul Erdős and Joel Spencer.

Probabilistic methods in combinatorics

.

Academic Press, 1973.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0308.05001

Zbl
0308.05001.

Title

probabilistic method

Canonical

name

ProbabilisticMethod

Date of creation

2013-03-22 13:05:53

Last modified on

2013-03-22 13:05:53

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

15

Author

bbukh (348)

Entry type

Topic

Classification

msc 05C80

Classification

msc 11K99

Related topic

GraphTheory

Related topic

ProofOfCrossingLemma

Related topic

RamseysTheorem2

Generated on Fri Feb 9 07:56:40 2018 by

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equivalence between the minor and topological minor of K_5 or K_{3,3}

equivalence between the minor and topological minor of

K

5

or

K

3

,

3

A graph

G

contains

K

5

or

K

3

,

3

as a minor iff it contains

K

5

or

K

3

,

3

as a

topological minor

(

http://planetmath.org/subdivision

). Where

K

5

is the

complete graph

of order 5 and

K

3

,

3

is the

complete bipartite graph

of order 6.

Remark that this

theorem

shows that Wagner’s theorem and Kuratowski’s theorem are

equivalent

.

Title

equivalence between the minor and topological minor of

K

5

or

K

3

,

3

Canonical name

EquivalenceBetweenTheMinorAndTopologicalMinorOfK5OrK33

Date of creation

2013-03-22 17:47:13

Last modified on

2013-03-22 17:47:13

Owner

jwaixs (18148)

Last modified by

jwaixs (18148)

Numerical id

8

Author

jwaixs (18148)

Entry type

Theorem

Classification

msc 05C83

Classification

msc 05C10

Generated on Fri Feb 9 07:56:45 2018 by

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The Annals of Having Compensated To Complete Surveys

The Annals of Having Compensated To Complete Surveys

The depth-first search (DFS) is a method for finding a

spanning

forest of a

connected graph

(specifically, a

spanning tree

in the case of an undirected connected graph), and is a useful for other . In particular, the DFS is useful for finding the

connected components

of a graph. This article currently describes only the DFS for an

undirected graph

.

\theoremstyle

definition

Definition

(DFS on an Undirected Graph)

Assume that the undirected graph

G

=

(

V

,

E

)

is

connected

. The DFS proceeds as follows.

1.

Fix a vertex

v

in

G

and mark it as

visited

.

2.

For each edge

(

v

,

w

)

from

v

to

w

:

–

If

w

has been visited, do nothing.

–

If

w

has not been visited, perform a DFS starting from

w

.

When the search is , it

G

into two sets. One the directed edges between visited vertices, called

tree edges

, which form a

directed spanning tree

of the graph. The other the remaining edges of

G

which were left unexamined because they were

incident

on a vertex that was already visited at some in the search. Because these unexamined edges can be seen as

returning

to vertices that were previously visited, they are called

back edges

.

References

1

Swamy, M.N.S., Thulasiraman, K.,

Graphs,

Networks

and Algorithms

, John Wiley and Sons, New York, 1981.

Title

The Annals of Having Compensated To

Complete

Surveys

Canonical name

TheAnnalsOfHavingCompensatedToCompleteSurveys

Date of creation

2013-11-27 10:59:44

Last modified on

2013-11-27 10:59:44

Owner

jacou (1000048)

Last modified by

(0)

Numerical id

18

Author

jacou (0)

Entry type

Algorithm

Classification

msc 05C85

Synonym

DFS

Related topic

GraphTheory

Related topic

SpanningTree

Related topic

ConnectedComponents

Generated on Fri Feb 9 07:56:46 2018 by

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