algorithmic complexity of weakly semiregular partitioning and the representation number abstract a graph g is weakly semiregular if there are two numbers a b such that the degree of every vertex is a or b the weakly semiregular number of a graph g denoted by w r g is the minimum number of subsets into which the edge set of g can be partitioned so that the subgraph induced by each subset is a weakly semiregular graph we present a polynomial time algorithm to determine whether the weakly semiregular number of a given tree is two on the other hand we show that determining whether w r g 2 for a given bipartite graph g with at most three numbers in its degree set is np complete among other results for every tree t we show that w r t 2 log 2 t o 1 where t denotes the maximum degree of t a graph g is a d d s graph if the degree of every vertex of g lies in the interval d d s a d d 1 graph is said to be semiregular the semiregular number of a graph g denoted by s r g is the minimum number of subsets into which the edge set of g can be partitioned so that the subgraph induced by each subset is a semiregular graph we prove that the semiregular number of a tree t is t 2 on the other hand we show that determining whether s r g 2 for a given bipartite graph g with g 6 is np complete in the second part of the work we consider the representation number a graph g has a representation modulo r if there exists an injective map l v g z r such that vertices v and u are adjacent if and only if l u l v is relatively prime to r the representation number denoted by r e p g is the smallest r such that g has a representation modulo r narayan and urick conjectured that the determination of r e p g for an arbitrary graph g is a difficult problem 38 in this work we confirm this conjecture and show that if n p p then for any 0 there is no polynomial time 1 n 2 approximation algorithm for the computation of representation number of regular graphs with n vertices

chromatic index of graphs with no cycle with a unique chord the class c of graphs that do not contain a cycle with a unique chord was recently studied by trotignon and vuskovic in press 23 who proved for these graphs strong structure results which led to solving the recognition and vertex colouring problems in polynomial time in the present paper we investigate how these structure results can be applied to solve the edge colouring problem in the class we give computational complexity results for the edge colouring problem restricted to c and to the subclass c composed of the graphs of c that do not have a 4 hole we show that it is np complete to determine whether the chromatic index of a graph is equal to its maximum degree when the input is restricted to regular graphs of c with fixed degree d 3 for the subclass c we establish a dichotomy if the maximum degree is d 3 the edge colouring problem is np complete whereas if d 3 the only graphs for which the chromatic index exceeds the maximum degree are the odd holes and the odd order complete graphs a characterization that solves edge colouring problem in polynomial time we determine two subclasses of graphs in c of maximum degree 3 for which edge colouring is polynomial finally we remark that a consequence of one of our proofs is that edge colouring in np complete for r regular tripartite graphs of degree d 3 for r 3

parameterized complexity of max lifetime target coverage in wireless sensor networks max lifetime target coverage can be viewed as a family of problems where the task is to partition the sensors into groups and assign their time slots such that the coverage lifetime is maximized while satisfying some coverage requirement unfortunately these problems are np hard to gain insight into the source of the complexity we initiate a systematic parameterized complexity study of two types of max lifetime target coverage max min target coverage and max individual target coverage we first prove that both problems remain np hard even in the special cases where each target is covered by at most two sensors or each sensor can cover at most two targets by contrast restricting the number of targets reduces the complexity of the considered problems in other words they are both fixed parameter tractable fpt with respect to the parameter number of targets moreover we extend our studies to the structural parameter number k of sensors covering at least two targets positively both problems are in fpt with respect to k finally we show that max min target coverage is in fpt with respect to the combined parameters number of groups and number of targets covered by each group

the complexity of lsh feasibility in this paper we study the complexity of the following feasibility problem given an n n similarity matrix s as input is there a locality sensitive hash lsh for s we show that the lsh feasibility problem is np hard even in the following strong promise version either s admits an lsh or s is at l1 distance at least n2 from every similarity that admits an lsh we complement this hardness result by providing an o 3n algorithm for the lsh feasibility problem which improves upon the naive n n time algorithm we prove that this running time is tight modulo constants under the exponential time hypothesis

the parameterised complexity of list problems on graphs of bounded treewidth we consider the parameterised complexity of several list problems on graphs with parameter treewidth or pathwidth in particular we show that list edge chromatic number and list total chromatic number are fixed parameter tractable parameterised by treewidth whereas list hamilton path is w1 hard even parameterised by pathwidth these results resolve two open questions of fellows fomin lokshtanov rosamond saurabh szeider and thomassen 2011

local edge colouring of yao like subgraphs of unit disk graphs the focus of the present paper is on providing a local deterministic algorithm for colouring the edges of yao like subgraphs of unit disk graphs these are geometric graphs such that for some positive integers l k the following property holds at each node v if we partition the unit circle centered at v into 2k equally sized wedges then each wedge can contain at most l points different from v we assume that the nodes are location aware i e they know their cartesian coordinates in the plane the algorithm presented is local in the sense that each node can receive information emanating only from nodes which are at most a constant depending on k and l but not on the size of the graph number of hops measured in the original underlying unit disk graph away from it and hence the algorithm terminates in a constant number of steps the number of colours used is 2kl 1 and this is optimal for local algorithms since the maximal degree is 2kl and a colouring with 2kl colours can only be constructed by a global algorithm thus showing that in this class of graphs the price for locality is only one additional colour

np hard graph problems and boundary classes of graphs any graph problem which is np hard in general graphs becomes polynomial time solvable when restricted to graphs in special classes when does a difficult problem become easy to answer this question we study the notion of boundary classes in the present paper we define this notion in its most general form describe several approaches to identify boundary classes and apply them to various graph problems

weighted and locally bounded list colorings in split graphs cographs and partial k trees abstract for a fixed number of colors we show that in node weighted split graphs cographs and graphs of bounded tree width one can determine in polynomial time whether a proper list coloring of the vertices of a graph such that the total weight of vertices of each color equals a given value in each part of a fixed partition of the vertices exists we also show that this result is tight in some sense by providing hardness results for the cases where any one of the assumptions does not hold the edge coloring variant is also studied as well as special cases of cographs and split graphs

polyhedron of triangle free simple 2 matchings in subcubic graphs a simple 2 matching in an edge weighted graph is a subgraph all of whose nodes have degree 0 1 or 2 we consider the problem of finding a maximum weight simple 2 matching that contains no triangles which is closely related to a class of relaxations of the traveling salesman problem tsp our main results are for graphs with maximum degree 3 a complete description of the convex hull of incidence vectors of triangle free simple 2 matchings and a strongly polynomial time algorithm for the above problem our system requires the use of a type of comb inequality introduced by grotschel and padberg for the tsp polytope that has 0 1 2 coefficients and hence is more general than the well known blossom inequality used in edmonds characterization of the simple 2 matching polytope

p k coloring problems in line graphs the p k coloring problems generalize the usual coloring problem by replacing stable sets by cliques and stable sets complexities of some variations of p k coloring problems split coloring and cocoloring are studied in line graphs polynomial algorithms or proofs of np completeness are given according to the complexity status we show that the most general p k coloring problems are more difficult than the cocoloring and the split coloring problems while there is no such relation between the last two problems we also give complexity results for the problem of finding a maximum p k colorable subgraph in line graphs finally upper bounds on the optimal values are derived in general graphs by sequential algorithms based on welsh powell and matula orderings

integer round up property for the chromatic number of some h perfect graphs we prove that every h perfect line graph and every t perfect claw free graph g has the integer round up property for the chromatic number for every non negative integral weight function c on the vertices of g the weighted chromatic number of g c can be obtained by rounding up its fractional relaxation as a corollary we obtain that the weighted chromatic number can be computed in polynomial time for these graphs finally we show a new example of a graph operation which preserves the integer round up property for the chromatic number and use it to provide a first example of a t perfect 3 colorable graph which does not have this property

jump number maximization for proper interval graphs and series parallel graphs in this paper the order theoretic problem of finding linear extensions of a partially ordered set poset with certain properties is mapped to the corresponding graphtheorectic problem given a poset a jump exists between two consecutive symbols in a topological sequence if there is no relation between these two symbols in the poset we want to find a topological sequence that minimizes or maximizes the total number of jumps in graph theoretic terms define the jump number optimization problem as finding a linear ordering of vertex sequence of a graph such that the number of pairs of consecutive vertices that are adjacent is maximized jump number minimization problem or minimized jump number maximization problem the complexities of both problems with respect to some special graphs are investigated first in this paper we then study the jump maximization problem for proper interval graphs and series parallel graphs from this study we hope that more researches can be derived from these interdisciplinary knowledge domains

tests for regulability of 2 d systems regulability tests of 2 d systems are transformed into stability tests of one variable polynomials in this paper the method can be used in regulability tests of multi dimensional systems without any substantial modification

space efficient euler partition and bipartite edge coloring abstract we describe space efficient algorithms for two problems on undirected multigraphs euler partition partitioning the edges into a minimum number of trails and bipartite edge coloring coloring the edges of a bipartite multigraph with the minimum number of colors let n m and 1 be the numbers of vertices and of edges and the maximum degree respectively of the input multigraph for euler partition we reduce the amount of working memory needed by a logarithmic factor from n m log n m bits to o n m bits while preserving a running time of o n m for bipartite edge coloring still using o n m bits of working memory we achieve a running time of o n m min log log log m log this is o m log log if m n log n log log n log n to be compared with o m log for the fastest known algorithm instrumental in obtaining the latter result is a new data structure presented here the subgraph stack that allows graph algorithms centered around recursive calls on substantially smaller subgraphs to be implemented in a space efficient manner

on the coefficient assignment of two dimensional systems in this note we prove that in two dimensional 2 d systems the arbitrary coefficient assignment cannot be achieved by using causal output feedbacks the problem however can be transformed into the design of systems over rings in order to find conditions guaranteeing the existence of a solution some related problems are also discussed

structure in approximation classes the study of the approximability properties of np hard optimization problems has recently made great advances mainly due to the results obtained in the field of proof checking the last important breakthrough proves the apx completeness of several important optimization problems and thus reconciles two distinct views of approximation classes syntactic and computational s khanna et al in proc 35th ieee symp on foundations of computer science ieee computer society press los alamitos ca 1994 pp 819 830 in this paper we obtain new results on the structure of several computationally defined approximation classes in particular after defining a new approximation preserving reducibility to be used for as many approximation classes as possible we give the first examples of natural npo complete problems and the first examples of natural apx intermediate problems moreover we state new connections between the approximability properties and the query complexity of npo problems

decompositions for the edge colouring of reduced indifference graphs the chromatic index problem finding the minimum number of colours required for colouring the edges of a graph is still unsolved for indifference graphs whose vertices can be linearly ordered so that the vertices contained in the same maximal clique are consecutive in this order we present new positive evidence for the conjecture every non neighbourhood overfull indifference graph can be edge coloured with maximum degree colours two adjacent vertices are twins if they belong to the same maximal cliques a graph is reduced if it contains no pair of twin vertices a graph is overfull if the total number of edges is greater than the product of the maximum degree by n 2 where n is the number of vertices we give a structural characterization for neighbourhood overfull indifference graphs proving that a reduced indifference graph cannot be neighbourhood overfull we show that the chromatic index for all reduced indifference graphs is the maximum degree we present two decomposition methods for edge colouring reduced indifference graphs with maximum degree colours

an upper bound for the choice number of star edge coloring of graphs abstract the star chromatic index of a multigraph g denoted s g is the minimum number of colors needed to properly color the edges of g such that no path or cycle of length four is bi colored a multigraph g is star k edge colorable if s g k dvo ak et al 2013 proved that every subcubic multigraph is star 7 edge colorable they conjectured in the same paper that every subcubic multigraph should be star 6 edge colorable in this paper we consider this problem in a more general setting we investigate star list edge coloring of general graph g and obtain an upper bound for the choice number of star edge coloring of graphs namely we proved that s l 2 3 2 1 2 1 2 2

when hamming meets euclid the approximability of geometric tsp and steiner tree we prove that the traveling salesman problem sc min tsp is sf max snp hard and thus sf np hard to approximate within some constant r 1 even if all cities lie in a euclidean space of dimension log n n is the number of cities and distances are computed with respect to any lp norm the running time of recent approximation schemes for geometric sc min tsp is doubly exponential in the number of dimensions our result implies that this dependence is necessary unless np has subexponential algorithms as an intermediate step we also prove the hardness of approximating sc min tsp in hamming spaces finally we prove a similar but weaker inapproximability result for the steiner minimal tree problem sc min st the reduction for sc min tsp uses error correcting codes the reduction for sc min st uses the integrality property of sc min cut linear programming relaxations the only previous inapproximability results for metric sc min tsp involved metrics where all distances are 1 or 2

bipartite edge coloring in o delta m time we show that a minimum edge coloring of a bipartite graph can be found in o delta m time where delta and m denote the maximum degree and the number of edges of g respectively it is equivalent to finding a perfect matching in a k regular bipartite graph in o km time by sharpening the methods a minimum edge coloring of a bipartite graph can be found in o p max delta log delta m time where p max delta is the largest prime factor of delta moreover a perfect matching in a k regular bipartite graph can be found in o pmax k m time

the 3 colorability problem on graphs with maximum degree four the 3 colorability problem is known to be np complete in the class of graphs with maximum degree four on the other hand due to the celebrated theorem of brooks the problem has a polynomial time solution for graphs with maximum degree three to make the complexity gap more precise we study a family of intermediate graph classes between these two extremes and classify all of them according to the computational complexity of the problem in particular we generalize brooks s theorem in the case of 3 colorability to a larger class by showing that every connected graph in that class is 3 colorable unless it is a complete graph on four vertices

approximating the max edge coloring problem the max edge coloring problem is a natural weighted generalization of the classical edge coloring problem arising in the domain of communication systems in this problem each color class is assigned the weight of the heaviest edge in this class and the objective is to find a proper edge coloring of the input graph minimizing the sum of all color classes weights we present new approximation results that improve substantially the known ones for several variants of the problem with respect to the class of the underlying graph in particular we deal with variants which are known to be np hard general and bipartite graphs or are proven to be np hard in this paper complete graphs with bi valued edge weights or whose complexity question still remains open trees

new algorithms for maximum disjoint paths based on tree likeness we study the classical mathsf np hard problems of finding maximum size subsets from given sets of k terminal pairs that can be routed via edge disjoint paths maxedp or node disjoint paths maxndp in a given graph the approximability of maxedp maxndp is currently not well understood the best known lower bound is 2 varomega sqrt log n assuming mathsf np not subseteq mathsf dtime n mathcal o log n this constitutes a significant gap to the best known approximation upper bound of mathcal o sqrt n due to chekuri et al theory comput 2 137 146 2006 and closing this gap is currently one of the big open problems in approximation algorithms in their seminal paper raghavan and thompson combinatorica 7 4 365 374 1987 introduce the technique of randomized rounding for lps their technique gives an mathcal o 1 approximation when edges or nodes may be used by mathcal o left log n log log n right paths in this paper we strengthen the fundamental results above we provide new bounds formulated in terms of the feedback vertex set number r of a graph which measures its vertex deletion distance to a forest in particular we obtain the following results for maxedp we give an mathcal o sqrt r log k r approximation algorithm up to a logarithmic factor our result strengthens the best known ratio mathcal o sqrt n due to chekuri et al as r le n further we show how to route varomega text opt pairs with congestion bounded by mathcal o log kr log log kr strengthening the bound obtained by the classic approach of raghavan and thompson for maxndp we give an algorithm that gives the optimal answer in time k r mathcal o r cdot n this is a substantial improvement on the run time of 2 kr mathcal o r cdot n which can be obtained via an algorithm by scheffler we complement these positive results by proving that maxedp is mathsf np hard even for r 1 and maxndp is mathsf w 1 hard when r is the parameter this shows that neither problem is fixed parameter tractable in r unless mathsf fpt mathsf w 1 and that our approximability results are relevant even for very small constant values of r

maximum cardinality simple 2 matchings in subcubic graphs a simple 2 matching in a graph is a subset of edges m such that every node is incident with at most two edges in m a simple 2 matching is called ck restricted k 2 if it contains no cycles of length k or less we begin by considering the problem of finding maximum cardinality simple 2 matchings in subcubic graphs i e graphs whose nodes have degree at most 3 for this problem we present a min max theorem a simple polynomial time algorithm and some other results then by generalizing an approach due to vornberger we show the following for subcubic graphs the maximum cardinality c3 and c4 restricted simple 2 matching problems are polynomial time solvable there exist min max theorems for these problems and for k 5 the maximum cardinality ck restricted simple 2 matching problems are np hard

the complexity of controlled selection we identify a class of problems called controlled selection problems and study their complexity a controlled selection problem involves a random selection of a sample from a large set subject to certain structural constraints the structural constraints model some desired relationships among the elements of the set which we want to preserve under random selection for example the set may be divided into equivalence classes and we may want to ensure that not too many elements are chosen from the same class or the set may correspond to vertices of a graph and we may want to ensure that the sample does not induce a cycle controlled selection problems arise in many disciplines including statistics mathematical programming combinatorial algorithms and resource allocation this is the first attempt to unify them we develop techniques for determining the complexity of controlled selection problems given the structural constraints we use these techniques to determine the complexity of many types of controlled selection problems we show that a controlled selection problem is polynomial time reducible to the problem of finding a maximum weight sample satisfying the structural constraints we also show that a general reduction in the opposite direction would imply that one way functions do not exist in contrast we show that most natural controlled selection problems have the same complexity within polynomial time as the problem of finding a maximum weight sample satisfying the structural constraints we also present efficient algorithms for several controlled selection problems

the tournament scheduling problem with absences abstract we study time scheduling problems with allowed absences as a new kind of graph coloring problem one may think of a sport tournament where each player each team is permitted a certain number t of absences we then examine how many rounds are needed to schedule the whole tournament in the worst case this upper limit depends on t and on the structure of the graph g whose edges represent the games that have to be played but also on whether or not the absences are announced before the tournament starts therefore we actually have two upper limits for the number of required rounds we have t g for pre scheduling if all absences are pre fixed and we have ol t g for on line scheduling if we have to stay flexible and deal with absences when they occur we conjecture that t g g 2 t and that ol t g g 2 t the first conjecture is stronger than the total coloring conjecture while the second is weaker than the on line list edge coloring conjecture our conjectures hold for all bipartite graphs for complete graphs we prove them partially lower and upper bounds to t g and ol t g for general multigraphs g are established too

local algorithms for edge colorings in udgs in this paper we consider two problems the edge coloring and the strong edge coloring problems on unit disk graphs udgs both problems have important applications in wireless sensor networks as they can be used to model link scheduling problems in such networks it is well known that both problems are np complete and approximation algorithms for them have been extensively studied under the centralized model of computation centralized algorithms however are not suitable for ad hoc wireless sensor networks whose devices typically have limited resources and lack the centralized coordination we develop local distributed approximation algorithms for the edge coloring and the strong edge coloring problems on unit disk graphs for the edge coloring problem our local distributed algorithm has approximation ratio 2 and locality 50 for the strong edge coloring problem on udgs we present two local distributed algorithms with different tradeoffs between their approximation ratio and locality the first algorithm has ratio 128 and locality 22 whereas the second algorithm has ratio 10 and locality 180

bounding the maximum size of a packet radio network the author addresses the problem arising in the topology design of packet radio networks prns which use time division multiplexing and have a diameter constraint of what is the maximum number n sub c f k of users which can be contained in a diameter k prn with f time slots per frame the author assumed that users cannot transmit and receive simultaneously and cannot transmit receive more than one packet at a time this assumption implies that no two channels accessed by the same user may be assigned the same time slot it is shown that the problem of determining n sub c f k is identical to the problem of determining the largest number of vertices which can be contained in an f edge colorable directed graph with diameter k lower bounds on n sub c f k for f 2 k 3 4 5 are obtained by generating large graphs of this type the graphs are constructed and colored by simple and fast procedures which are similar for different values of f and k an extensive bibliography on the edge coloring problem is included

the quaternion based three dimensional beam theory this paper presents the equations for the implementation of rotational quaternions in the geometrically exact three dimensional beam theory a new finite element formulation is proposed in which the rotational quaternions are used for parametrization of rotations along the length of the beam the formulation also satisfies the consistency condition that the equilibrium and the constitutive internal force and moment vectors are equal in its weak form a strict use of the quaternion algebra in the derivation of governing equations and for the numerical solution is presented several numerical examples demonstrate the validity performance and accuracy of the proposed approach c 2009 elsevier b v all rights reserved

link scheduling in polynomial time two polynomial time algorithms are given for scheduling conversations in a spread spectrum radio network the constraint on conversations is that each station can converse with only one other station at a time the first algorithm is strongly polynomial and finds a schedule of minimum length that allows each pair of neighboring stations to converse directly for a prescribed length of time the second algorithm is designed for the situation in which messages must be relayed multiple hops the algorithm produces in polynomial time a routing vector and compatible link schedule that jointly meet a prespecified end to end demand so that the schedule has the smallest possible length

the complexity of path coloring and call scheduling abstract modern high performance communication networks pose a number of challenging problems concerning the efficient allocation of resources to connection requests in all optical networks with wavelength division multiplexing connection requests must be assigned paths and colors wavelengths such that intersecting paths receive different colors and the goal is to minimize the number of colors used this path coloring problem is proved np hard for undirected and bidirected ring networks path coloring in undirected tree networks is shown to be equivalent to edge coloring of multigraphs which implies a polynomial time optimal algorithm for trees of constant degree as well as np hardness and an approximation algorithm with absolute approximation ratio 4 3 and asymptotic approximation ratio 1 1 for trees of arbitrary degree for bidirected trees path coloring is shown to be np hard even in the binary case a polynomial time optimal algorithm is given for path coloring in undirected or bidirected trees with n nodes under the assumption that the number of paths touching every single node of the tree is o log n 1 e call scheduling is the problem of assigning paths and starting times to calls in a network with bandwidth reservation such that the maximum completion time is minimized in the case of unit bandwidth requirements unit edge capacities and unit call durations call scheduling is equivalent to path coloring if either the bandwidth requirements or the call durations can be arbitrary call scheduling is shown np hard for virtually every network topology

the sylvester resultant with applications to systems over rings the sylvester resultant is studied in this paper for systems over rings the systems over rings may have zero coprime fractions fractor coprime fractions and fraction with common factors the sylvester resultant can be used to test these situations it can also be used to achieve the design of coefficient assignment in the first case to test the regulability in the second case and to find the fractor coprime fractions in the third case

application of the absolute nodal co ordinate formulation to multibody system dynamics abstract the floating frame of reference formulation is currently the most widely used approach in flexible multibody simulations the use of this approach however has been limited to small deformation problems in this investigation the computer implementation of the newabsolute nodal co ordinate formulationand its use in the small and large deformation analysis of flexible multibody systems that consist of interconnected bodies are discussed while in the floating frame of reference formulation a mixed set of absolute reference and local elastic co ordinates are used in the absolute nodal co ordinate formulation only absolute co ordinates are used in the absolute nodal co ordinate formulation new interpretation of the nodal co ordinates of the finite elements is used no infinitesimal or finite rotations are used as nodal co ordinates from beams and plates instead global slopes are used to define the element nodal co ordinates using this interpretation of the element co ordinates beams and plates can be considered as isoparametric elements and as a result exact modelling of the rigid body dynamics can be obtained using the element shape function and the absolute nodal co ordinates unlike the floating frame of reference approach no co ordinate transformation is required in order to determine the element inertia the mass matrix of the finite elements is a constant matrix and therefore the centrifugal and coriolis forces are equal to zero when the absolute nodal co ordinate formulation is used another advantage of using the absolute nodal co ordinate formulation in the dynamic simulation of multibody systems is its simplicity in imposing some of the joint constraints and also its simplicity in formulating the generalized forces due to spring damper elements the results obtained in this investigation show an excellent agreement with the results obtained using the floating frame of reference formulation when large rotation small deformation problems are considered

the np completeness of edge coloring we show that it is np complete to determine the chromatic index of an arbitrary graph the problem remains np complete even for cubic graphs

development of simple models for the elastic forces in the absolute nodal co ordinate formulation abstract the objective of this study is to develop simple and accurate elastic force models that can be used in the absolute nodal co ordinate formulation for the analysis of two dimensional beams these force models which account for the coupling between bending and axial deformations are derived using a continuum mechanics approach without the need for introducing a local element co ordinate system four new different force models that include different degrees of complexity are presented it is shown that the vector of the elastic forces can be significantly simplified as compared to the elastic force model developed for the absolute nodal co ordinate formulation using a local element frame 1 despite the simplicity of the new models they account for elastic non linearity in the strain displacement relationship therefore they lead to more accurate results as compared to the more complex models developed using the local frame method which does not account for the non linearities in the strain displacement relationships numerical results are presented in order to demonstrate the use of the new models and test their performances in the analysis of large deformations of flexible multibody systems

on excessive index of certain networks matching is one of the most extensively studied areas in computer science and is interesting from the combinatorial point of view as well a matching in a graph g v e is a subset m of edges no two of which have a vertex in common a matching m is said to be perfect if every vertex in g is an endpoint of one of the edges in m the excessive index of a graph g is the minimum number of perfect matchings to cover the edge set of g the study of excessive index has a number of applications particularly in scheduling theory in this paper we determine the excessive index for certain classes of graphs including augmented butterfly network and honeycomb network we also prove that the excessive index does not exist for butterfly and benes networks

minimizing makespan in no wait job shops in this paper we study polynomial time approximation schemes ptases for the no wait job shop scheduling problem with the makespan objective function it is known that the problem is maxsnp hard in the case when each job is allowed to have three operations or more we show that if each job has at most two operations the problem admits a ptas if the number of machines is a constant i e not part of the input if the number of machines is not a constant we show that the problem is hard to approximate within a factor better than 5 4

algorithmic complexity of proper labeling problems a proper labeling of a graph is an assignment of integers to some elements of a graph which may be the vertices the edges or both of them such that we obtain a proper vertex coloring via the labeling subject to some conditions the problem of proper labeling offers many variants and received a great interest during recent years we consider the algorithmic complexity of some variants of the proper labeling problems we present some polynomial time algorithms and np completeness results for them

chromatic gallai identities operating on lovasz number if g is a triangle free graph then two gallai identities can be written as alpha g overline chi l g v g alpha l g overline chi g where alpha and overline chi denote the stability number and the clique partition number and l g is the line graph of g we show that surprisingly both equalities can be preserved for any graph g by deleting the edges of the line graph corresponding to simplicial pairs of adjacent arcs according to any acyclic orientation of g as a consequence one obtains an operator phi which associates to any graph parameter beta such that alpha g le beta g le overline chi g for all graph g a graph parameter phi beta such that alpha g le phi beta g le overline chi g for all graph g we prove that vartheta g le phi vartheta g and that phi overline chi f g le overline chi f g for all graph g where vartheta is lovasz theta function and overline chi f is the fractional clique partition number moreover overline chi f g le phi vartheta g for triangle free g comparing to the previous strengthenings psi vartheta and vartheta triangle of vartheta numerical experiments show that phi vartheta is a significant better lower bound for overline chi than vartheta

densities matchings and fractional edge colorings given a multigraph g v e with a positive rational weight w e on each edge e the weighted density problem wdp is to find a subset u of v with u ge 3 and odd that maximizes 2w

edge colouring of regular graphs of large degree we consider the following conjecture letgbe ak regular simple graph with an even numbernof vertices ifk n 2thengisk edge colourable we show that this conjecture is true for graphs that are join of two graphs and we provide a polynomial time algorithm for finding a k edge colouring of these graphs

computational complexity of covering three vertex multigraphs a covering projection from a graph g onto a graph h is a mapping of the vertices of g onto the vertices of h such that for every vertex v of g the neighborhood of v is mapped bijectively onto the neighborhood of its image moreover if g and h are multigraphs then this local bijection has to preserve multiplicities of the neighbors as well the notion of covering projection stems from topology but has found applications in areas such as the theory of local computation and construction of highly symmetric graphs it provides a restrictive variant of the constraint satisfaction problem with additional symmetry constraints on the behavior of the homomorphisms of the structures involved we investigate the computational complexity of the problem of deciding the existence of a covering projection from an input graph g to a fixed target graph h among other partial results this problem has been shown np hard for simple regular graphs h of valency greater than 2 and a full characterization of computational complexity has been shown for target multigraphs with 2 vertices we extend the previously known results to the ternary case i e we give a full characterization of the computational complexity in the case of multigraphs with 3 vertices we show that even in this case a p np completeness dichotomy holds

algorithmic problems in right angled artin groups complexity and applications abstract in this paper we consider several classical and novel algorithmic problems for right angled artin groups some of which are closely related to graph theoretic problems and study their computational complexity we study these problems with a view towards applications to cryptography

an approximation algorithm for diagnostic test scheduling in multicomputer systems the problem of diagnostic test scheduling dts is to assign to each edge e of a diagnostic graph g a time interval of length l e so that intervals corresponding to edges at any given vertex do not overlap and the overall finishing time is minimum in this correspondence we show that the dts problem is np complete then we present a longest first sequential scheduling algorithm which runs in worst case time o dm log n and uses o m space to produce a solution of length less than four times optimal then we show that the general performance bound can be strengthened to 3 opt g for low degree graphs and to 2 opt g in some special cases of binomial diagnostic graphs

satisfactory graph partition variants and generalizations the satisfactory partition problem asks for deciding if a given graph has a partition of its vertex set into two nonempty parts such that each vertex has at least as many neighbors in its part as in the other part this problem was introduced by gerber and kobler m gerber d kobler algorithmic approach to the satisfactory graph partitioning problem european journal of operational research 125 2000 283 291 and studied further by other authors in this paper we first review some applications and related problems then we survey structural complexity and approximation results obtained for satisfactory partition and for some of its variants and generalizations a list of open questions concludes this survey

the np completeness of chromatic index in triangle free graphs with maximum vertex of degree 3 we restrict the known case about the complexity of edge coloring for cubic graphs in the case of triangle free graphs with maximum vertex degree 3

computing clique and chromatic number of circular perfect graphs in polynomial time a main result of combinatorial optimization is that clique and chromatic number of a perfect graph are computable in polynomial time grotschel et al in combinatorica 1 2 169 197 1981 perfect graphs have the key property that clique and chromatic number coincide for all induced subgraphs we address the question whether the algorithmic results for perfect graphs can be extended to graph classes where the chromatic number of all members is bounded by the clique number plus one we consider a well studied superclass of perfect graphs satisfying this property the circular perfect graphs and show that for such graphs both clique and chromatic number are computable in polynomial time as well in addition we discuss the polynomial time computability of further graph parameters for certain subclasses of circular perfect graphs all the results strongly rely upon lovasz s theta function

on the correct representation of bending and axial deformation in the absolute nodal coordinate formulation with an elastic line approach abstract in finite element methods that are based on position and slope coordinates a representation of axial and bending deformation by means of an elastic line approach has become popular such beam and plate formulations based on the so called absolute nodal coordinate formulation have not yet been verified sufficiently enough with respect to analytical results or classical nonlinear rod theories examining the existing planar absolute nodal coordinate element which uses a curvature proportional bending strain expression it turns out that the deformation does not fully agree with the solution of the geometrically exact theory and even more serious the normal force is incorrect a correction based on the classical ideas of the extensible elastica and geometrically exact theories is applied and a consistent strain energy and bending moment relations are derived the strain energy of the solid finite element formulation of the absolute nodal coordinate beam is based on the st venant kirchhoff material therefore the strain energy is derived for the latter case and compared to classical nonlinear rod theories the error in the original absolute nodal coordinate formulation is documented by numerical examples the numerical example of a large deformation cantilever beam shows that the normal force is incorrect when using the previous approach while a perfect agreement between the absolute nodal coordinate formulation and the extensible elastica can be gained when applying the proposed modifications the numerical examples show a very good agreement of reference analytical and numerical solutions with the solutions of the proposed beam formulation for the case of large deformation pre curved static and dynamic problems including buckling and eigenvalue analysis the resulting beam formulation does not employ rotational degrees of freedom and therefore has advantages compared to classical beam elements regarding energy momentum conservation

h coloring degree bounded acyclic digraphs an np complete coloring or homomorphism problem may become polynomial time solvable when restricted to graphs with degrees bounded by a small number but remain np complete if the bound is higher we investigate an analogous phenomenon for digraphs focusing on the three smallest digraphs h with np complete h colorability problems it turns out that in all three cases the h coloring problem is polynomial time solvable for digraphs with in degrees at most 1 regardless of the out degree bound respectively with out degrees at most 1 regardless of the in degree bound on the other hand as soon as both in and out degrees are bounded by constants greater than or equal to 2 all three problems are again np complete interestingly the situation changes when we further restrict the inputs to be acyclic digraphs in two of the three cases the np complete problems remain np complete in the third case the h coloring problem turns out to be polynomial time solvable for acyclic digraphs with in degrees at most 2 regardless of the out degree bound respectively with out degrees at most 2 regardless of the in degree bound we also show that in this case as soon as both in and out degrees are bounded by constants greater than or equal to 3 the h coloring problem is once again np complete even for acyclic digraphs a conjecture proposed for graphs h by feder hell and huang states that any variant of the h coloring problem which is np complete without degree constraints is also np complete with degree constraints provided the degree bounds are high enough it has been verified for h coloring of digraphs but the degree bounds are quite high and are stated in terms of the sum of in and out degrees our results confirm the conjecture with the best possible bounds that are needed for np completeness moreover they underscore the fact that the bound must apply separately to both the in degrees and the out degrees

on edge colouring indifference graphs copyright c 1997 elsevier science b v all rights reserved vizingrs theorem states that the chromatic index g of a graph g is either the maximum degree d g or d g 1 a graph g is called overfull if ve g v d g vv g v 2 a sufficient condition for g d g 1 is that g contains an overfull subgraph h with d h d g plantholt proved that this condition is necessary for graphs with a universal vertex in this paper we conjecture that for indifference graphs this is also true as supporting evidence we prove this conjecture for general graphs with three maximal cliques and with no universal vertex and for indifference graphs with odd maximum degree for the latter subclass we prove that d

the strain based beam finite elements in multibody dynamics we present a strain based finite element formulation for the dynamic analysis of flexible elastic planar multibody systems composed of planar beams we consider finite displacements rotations and strains the discrete dynamic equations of motion are obtained by the collocation method the strains are the basic interpolated variables which makes the formulation different from other formulations the further speciality of the formulation is the strong satisfaction of the cross sectional constitutive conditions at collocation points in order to avoid the nested integrations a special algorithm for the numerical integration over the length of the finite element is proposed the midpoint scheme is used for the time integration the performance of the formulation is illustrated via numerical examples including a stiff multibody system