hardness of solving relational equations minimal solutions play a crucial role in describing all solutions of relational equations for this reason the problem of computing minimal solutions has for long been examined the literature contains several algorithms for computing minimal solutions recently contributions regarding computational complexity of the problem itself appeared the complexity aspect is clearly of fundamental importance however the existing results contain serious flaws in this paper we inspect the existing contributions clarify the flaws examine the problem of complexity of computing minimal solutions prove that there is no efficient algorithm computing all minimal solutions and discuss further ramifications of our observations

sup t norm and inf residuum are one type of relational product unifying framework and consequences we present a simple framework which enables us to consider the well known sup t norm and inf residuum products of relations as two particular cases of a single more general type of product we present basic properties of the framework and consequences for the theory of fuzzy relations informally the paper implies that in many cases of fuzzy relational modeling such as in solving fuzzy relational equations there is no need to develop the methods for sup t norm and inf residuum products separately because these methods are just two particular instances of a single method

on the relationship between nlc width and linear nlc width in this paper we consider nlc width nlct width and linear nlc width bounded graphs we show that the set of all complete binary trees has unbounded linear nlc width and that the set of all co graphs has unbounded nlct width since trees have nlct width 3 and co graphs have nlc width 1 it follows that the family of linear nlc width bounded graph classes is a proper subfamily of the family of nlct width bounded graph classes and that the family of nlct width bounded graph classes is a proper subfamily of the family of nlc width bounded graph classes

regularity for double phase variational problems we prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases according to the zero set of a modulating coefficient a cdot the model case is given by the functional w mapsto int dw p a x dw q rm d x where q p and a cdot geqq 0

restrictions of graph partition problems let j be a set of n jobs all with job processing time or job length one and let m be a set of k machines incompatibilities between jobs are described by an undirected graph g j e with vertex set j if g contains an edge between two jobs j and j j j we demand that these two jobs can not be executed by the same machine an assignment of the jobs to the machines which satisfies the incompatibility relation is called a schedule the processing time of a machine is given by the number of assigned jobs since each job has processing time one

characterizations for restricted graphs of nlc width 2 in this paper we give a finite forbidden subgraph characterization of graphs defined by nlc width 2 expressions by nlct width 2 expressions or by linear nlc width 2 expressions that have tree width 1

regularity results for a class of functionals with non standard growth we consider the integral functional under non standard growth assumptions that we call p x type namely we assume that a relevant model case being the functional under sharp assumptions on the continuous function p x 1 we prove regularity of minimizers energies exhibiting this growth appear in several models from mathematical physics

graphical methods for efficient likelihood inference in gaussian covariance models in graphical modelling a bi directed graph encodes marginal independences among random variables that are identified with the vertices of the graph we show how to transform a bi directed graph into a maximal ancestral graph that i represents the same independence structure as the original bi directed graph and ii minimizes the number of arrowheads among all ancestral graphs satisfying i here the number of arrowheads of an ancestral graph is the number of directed edges plus twice the number of bi directed edges in gaussian models this construction can be used for more efficient iterative maximization of the likelihood function and to determine when maximum likelihood estimates are equal to empirical counterparts

boundary regularity of minimizers of double phase functionals abstract in this paper we treat the functional of double phase with variable exponents d u g p x a x d u g q x d x where a x is a non negative holder continuous function with 0 1 p x and q x holder continuous functions with 1 p x q x p x and g i j g x i j 1 2 for a continuous positive definite matrix valued function g g we prove that the minimizer of the above functional with suitable dirichlet boundary condition is holder continuous up to the boundary when g is holder continuous we see also that du is locally holder continuous

minimal solutions of general fuzzy relation equations on linear carriers an algebraic characterization abstract this paper considers a general fuzzy relation equation which has minimal solutions if it is solvable in this case an algebraic characterization is introduced which provides an interesting method to compute minimal solutions in this general setting moreover a comparison with other frameworks is also given

resolution of fuzzy relational equations method algorithm and software with applications analytical methods and algorithms for inverse problem resolution of fuzzy linear systems of equations in some bl algebras godel algebra in case of max min and min max compositions and goguen algebra in case of max product composition are presented algorithms with software realization for solving fuzzy linear systems of equations are proposed applications include fuzzy optimization with fuzzy linear systems of equation constraint fuzzy machines and covering problem

induced subgraph isomorphism the complexity of the subgraph isomorphism problem where the pattern graph is of fixed size is well known to depend on the topology of the pattern graph here we present two results which in contrast provide evidence that no topology of an induced subgraph of fixed size can be substantially easier to detect or count than an independent set of related size we show that any fixed pattern graph having a maximum independent set of size k that is disjoint from other maximum independent sets is not easier to detect as an induced subgraph than an independent set of size k it follows in particular that an induced path on 2 k 1 vertices is not easier to detect than an independent set on k vertices and that an induced cycle on 2k vertices is not easier to detect than an independent set on k vertices in view of linear time upper bounds on the detection of induced path of length two and three our lower bound is tight similar corollaries hold for the detection of induced complete bipartite graphs and an induced paw and its generalizations we show also that for an arbitrary pattern graph h on k vertices with no isolated vertices there is a simple subdivision of h resulting from splitting each edge into a path of length four and attaching a distinct path of length three at each vertex of degree one that is not easier to detect or count than an independent set on k vertices respectively next we show that the so called diamond and its generalizations on k vertices are not easier to detect as induced subgraphs than an independent set on three vertices or an independent set on k vertices respectively for c 4 we give a weaker evidence of its hardness in terms of an independent set on three vertices finally we derive several results relating the complexity of the edge colored variant of induced subgraph isomorphism to that of the standard variant

recognizing circle graphs in polynomial time the main result of this paper is an 0 v x e time algorithm for deciding whether a given graph is a circle graph that is the intersection graph of a set of chords on a circle the algorithm utilizes two new graph theoretic results regarding necessary induced subgraphs of graphs having neither articulation points nor similar pairs of vertices furthermore as a substep of the algorithm it is shown how to find in 0 v x e time a decomposition of a graph into prime graphs thereby improving on a result of cunningham

on the double roman domination of graphs abstract a double roman dominating function of a graph g is a labeling f v g 0 1 2 3 such that if f v 0 then the vertex v must have at least two neighbors labeled 2 under f or one neighbor with f w 3 and if f v 1 then v must have at least one neighbor with f w 2 the double roman domination number dr g of g is the minimum value of v v g f v over such functions in this paper we firstly give some bounds of the double roman domination numbers of graphs with given minimum degree and graphs of diameter 2 and further we get that the double roman domination numbers of almost all graphs are at most n then we obtain sharp upper and lower bounds for d r g d r g moreover a linear time algorithm for the double roman domination number of a cograph is given and a characterization of the double roman cographs is provided those results partially answer two open problems posed by beeler et al 2016

unattainable solutions of a fuzzy relation equation in a fuzzy relation equation the properties of the solution set have been investigated especially when the basic spaces are all finite sets however when the basic spaces are infinite a few properties are known in this paper we use the concepts of attainability and unattainability of a solution to clarify some properties of the solution set and show the condition for existence of a partially attainable solution and an unattainable one

regularity results for stationary electro rheological fluids we prove regularity results for weak solutions to systems modelling electro rheological fluids in the stationary case as proposed in 27 31 a particular case of the system we consider is

a linear recognition algorithm for cographs cographs are the graphs formed from a single vertex under the closure of the operations of union and complement another characterization of cographs is that they are the undirected graphs with no induced paths on four vertices cographs arise naturally in such application areas as examination scheduling and automatic clustering of index terms furthermore it is known that cographs have a unique tree representation called a cotree using the cotree it is possible to design very fast polynomial time algorithms for problems which are intractable for graphs in general such problems include chromatic number clique determination clustering minimum weight domination isomorphism minimum fill in and hamiltonicity in this paper we present a linear time algorithm for recognizing cographs and constructing their cotree representation

partial regularity of p x harmonic maps let g x and hij u be uniformly elliptic symmetric matrices and assume that hij u and p x 2 are sufficiently smooth we prove partial regularity of minimizers for the functional f u z

incremental modular decomposition modular decomposition is a form of graph decomposition that has been discovered independently by researchers in graph theory game theory network theory and other areas this paper reduces the time needed to find the modular decomposition of a graph from o n 3 to o n 2 together with a new algorithm for transitive orientation given in 21 this leads to fast new algorithms for a number of problems in graph recognition and isomorphism including recognition of comparability graphs and permutation graphs the new algorithm works by inserting each vertex successively into the decomposition tree using o n time to insert each vertex

some properties of sup min fuzzy relational equations on infinite domains abstract this paper deals with a sup min fuzzy relational equation on infinite domains some properties of its solution set will be obtained in particular a necessary and sufficient condition describing the attainable solution resp the unattainable solution is formulated in the end the solution set will be given under some conditions

on switching classes nlc width cliquewidth and treewidth in this paper we consider a connection between switching of undirected graphs and the notions of nlc width cliquewidth and treewidth in particular we show that the nlc widths and the cliquewidths of two graphs in a switching class are at most a constant factor apart 2 for the former 4 for the latter a similar result can be shown not to hold for treewidth it is easy to find a switching classes in which the distance between the lowest treewidth and the highest is dependent on the number of vertices of the graph we also show that for nlc width every width between the lowest and the highest of the switching class is attained by some graph in that switching class we prove that this also holds for treewidth

strong triadic closure in cographs and graphs of low maximum degree abstract the maxstc problem is an assignment of the edges with two types of labels namely strong and weak that maximizes the number of strong edges such that any two vertices that have a common neighbor with a strong edge are adjacent the cluster deletion problem seeks for the minimum number of edge removals of a given graph such that the remaining graph is a disjoint union of cliques both problems are known to be np hard and an optimal solution for the cluster deletion problem provides a feasible solution for the maxstc problem however not necessarily an optimal one in this work we conduct the first systematic study that reveals graph families for which the optimal solutions for maxstc and cluster deletion coincide we first show that maxstc coincides with cluster deletion on cographs and thus maxstc is solvable in polynomial time on cographs as a side result we give an interesting computational characterization of the maximum independent set on the cartesian product of two cographs furthermore we address the influence of the low degree bounds to the complexity of the maxstc problem we show that this problem is polynomial time solvable on graphs of maximum degree three whereas maxstc becomes np complete on graphs of maximum degree four the proof of the latter result implies that there is no subexponential time algorithm for maxstc unless the exponential time hypothesis fails

on extended p 4 reducible and extended p 4 sparse graphs abstract a graph g was defined in 16 as p 4 reducible if no vertex in g belongs to more than one chordless path on four vertices or p 4 a graph g is defined in 15 as p 4 sparse if no set of five vertices induces more than one p 4 in g p 4 sparse graphs generalize both p 4 reducible and the well known class of p 4 free graphs or cographs in an extended abstract in 11 the first author introduced a method using the modular decomposition tree of a graph as the framework for the resolution of algorithmic problems this method was applied to the study of p 4 sparse and extended p 4 sparse graphs in this paper we begin by presenting the complete information about the method used in 11 we propose a unique tree representation of p 4 sparse and a unique tree representation of p 4 reducible graphs leading to a simple linear recognition algorithm for both classes of graphs in this way we simplify and unify the solutions for these problems presented in 16 19 the tree representation of an n vertex p 4 sparse or a p 4 reducible graph is the key for obtaining o n time algorithms for the weighted version of classical optimization problems solved in 20 these problems are np complete on general graphs finally by relaxing the restriction concerning the exclusion of the c 5 cycles from p 4 sparse and p 4 reducible graphs we introduce the class of the extended p 4 sparse and the class of the extended p 4 reducible graphs we then show that a minimal amount of additional work suffices for extending most of our algorithms to these new classes of graphs

vertex disjoint paths on clique width bounded graphs we show that the l vertex disjoint paths problem between l pairs of vertices can be solved in linear time for co graphs but is np complete for graphs of clique width at most 6 and nlc width at most 4 the np completeness follows from the fact that the line graph of a graph of tree width k has clique width at most 2k 2 and nlc width at most k 2 and a result by nishizeki et al the edge disjoint paths problem is np complete for series parallel graphs discrete appl math 115 2001 177 186 the vertex disjoint paths problem is the first graph problem shown to be np complete on graphs of bounded clique width but solvable in linear time on co graphs and graphs of bounded tree width additionally we show that the r vertex disjoint paths problem between each of l pairs of vertices can be solved in polynomial time for co graphs if l is given to the input and for graphs of bounded clique width if l is fixed

bivalent and other solutions of fuzzy relational equations via linguistic hedges we show that the well known results regarding solutions of fuzzy relational equations and their systems can easily be generalized to obtain criteria regarding constrained solutions such as solutions which are crisp relations when the constraint is empty constrained solutions are ordinary solutions the generalization is obtained by employing intensifying and relaxing linguistic hedges conceived in this paper as certain unary functions on the scale of truth degrees one aim of the paper is to highlight the problem of constrained solutions and to demonstrate that this problem naturally appears when identifying unknown relations the other is to emphasize the role of linguistic hedges as constraints

coupon coloring of cographs abstract coupon coloring is a new coloring which has many applications a k coupon coloring of a graph g is a k coloring of g by colors k 1 2 k such that the neighborhood of every vertex of g contains vertices of all colors from k the maximum integer k for which a k coupon coloring exists is called the coupon coloring number of g and it is denoted by c g in this paper we studied the coupon coloring of cographs which are graphs that can be generated from the single vertex graph k 1 by complementation and disjoint union and have applications in many interesting problems we use the cotree representation of a cograph to give a polynomial time algorithm to color the vertices of a cograph and then prove that this coloring is a coupon coloring with maximum colors hence get the coupon coloring numbers of the cograph

complexity of independent set reconfigurability problems we study problems of reconfigurability of independent sets in graphs we consider three different models token jumping token sliding and token addition and removal and analyze relationships between them we prove that independent set reconfigurability in perfect graphs under any of the three models generalizes the shortest path reconfigurability problem in general graphs and is therefore pspace complete on the positive side we give polynomial results for even hole free graphs and p 4 free graphs

regularity and existence of solutions of elliptic equations with p q growth conditions for some positive constants m a4 and for exponents q 2 p 2 2 under 1 2 1 3 and some other assumptions by assuming also that the quotient q p is sufficiently close to one in dependence on n precisely if q p c n n 2 then we prove that every weak solution to 1 1 of class 4 4 0 is locally lipschitz continuous in q moreover there are positive constants 8 c and 0 2 1 such that

minimal solutions of generalized fuzzy relational equations probabilistic algorithm based on greedy approach abstract the paper deals with generalized fuzzy relational equations that are defined within a recently introduced framework of sup preserving aggregation structures generalized fuzzy relational equations subsume all previously studied types of fuzzy relational equations namely those that are based on sup t norm and inf residuum compositions the paper contributes to previous studies of generalized fuzzy relational equations by presenting a method for constructing all minimal solutions and consequently for determining the whole solution set for any given generalized fuzzy relational equation that is solvable and for which every solution is bounded from below by a minimal solution moreover in the paper we present a simple probabilistic algorithm for finding all minimal solutions

on interior regularity of minimizers of p x energy functionals abstract the aim of the paper is to obtain everywhere c 1 regularity in the interior for a minimizer of the p x energy functional e v g x h i j v d v i x d v j x p x 2 d x defined for maps v r m r n under the so called one sided condition supposing the lipschitz continuity assumption on the exponent p x 2

on nested split graphs whose second largest eigenvalue is less than 1 abstract we determine all nested split graphs i e graphs having no induced subgraphs equal to 2 k 2 p 4 or c 4 having the second largest eigenvalue less than 1 and give some data and observations regarding obtained results

an optimal path cover algorithm for cographs abstract the class of cographs or complement reducible graphs arises naturally in many different areas of applied mathematics and computer science in this paper we present an optimal algorithm for determining a minimum path cover for a cograph g in case g has a hamiltonian path cycle our algorithm exhibits the path cycle as well

on clique separators nearly chordal graphs and the maximum weight stable set problem clique separators in graphs are a helpful tool used by tarjan as a divide and conquer approach for solving various graph problems such as the maximum weight stable set mws problem maximum clique graph coloring and minimum fill in but few examples of graph classes having clique separators are known we use this method to solve mws in polynomial time for two classes where the unweighted maximum stable set ms problem is solvable in polynomial time by augmenting techniques but the complexity of the mws problem was open another example namely a result by alekseev for the mws problem on a subclass of p 5 free graphs obtained by clique separators can be improved by our techniques we also combine clique separators with decomposition by homogeneous sets in graphs and use the following notion a graph is email protected if for each of its vertices the subgraph induced by the set of its nonneighbors has property p we deal with the cases email protected chordal perfect this also simplifies a result obtained by a method called struction

on line algorithms for orders partially ordered sets appear as a basic tool in computer science and are particularly accurate for modeling dynamic behavior of complex systems motivated by considerations on the diagnosis of distributed computations a new kind of algorithm on posets has been developed and is now widely considered in this paper we present this on line algorithmics on posets and we survey the main results obtained under this approach

on algorithms for p 5 gem free graphs a graph is p5 gem free when it does not contain p5 an induced path with five vertices or a gem a graph formed by making an universal vertex adjacent to each of the four vertices of the induced path p4 as an induced subgraph we present o n2 time recognition algorithms for chordal gem free graphs and for p5 gem free graphs using a characterization of p5 gem free graphs by their prime graphs with respect to modular decomposition and their modular decomposition trees a brandstadt d kratsch on the structure of p5 gem free graphs discrete appl math 145 2005 155 166 we give linear time algorithms for the following np complete problems on p5 gem free graphs minimum coloring maximum weight stable set maximum weight clique and minimum clique cover

a linear time recognition algorithm for p 4 reducible graphs p4 reducible graphs are precisely the graphs none of whose vertices belong to more than one chordless path with three edges as it turns out the class of p4 reducible graphs strictly contains the well known class of cographs a remarkable property of p4 reducible graphs is their unique tree representation up to isomorphism in this paper we present a linear time algorithm to recognize p4 reducible graphs and to construct their corresponding tree representation

leximax minimum solution of addition min fuzzy relation inequalities abstract addition min fuzzy relation inequalities are introduced to characterize the peer to peer p2p network system in order to decrease the network congestion considering both efficiency and fairness we define the leximax minimum solution for system of addition min fuzzy relation inequalities the leximax optimality refines both pareto optimality and max optimality for solving the leximax minimum solution concepts of minimum zero point and minimax value are defined with corresponding resolution algorithms i e algorithms i and ii based on these two algorithms algorithm iii is further developed to find the unique leximax minimum solution numerical examples illustrate the validity of our proposed algorithms

on the relationship between clique width and treewidth treewidth is generally regarded as one of the most useful parameterizations of a graph s construction clique width is a similar parameterization that shares one of the powerful properties of treewidth namely if a graph is of bounded treewidth or clique width then there is a polynomial time algorithm for any graph problem expressible in monadic second order logic using quantifiers on vertices in the case of clique width you must assume a clique width parse expression is given in studying the relationship between treewidth and clique width courcelle and olariu discrete appl math 101 2000 pp 77 114 showed that any graph of bounded treewidth is also of bounded clique width in particular for any graph g with treewidth k the clique width of g is at most 4 2k 1 1 in this paper we improve this result by showing that the clique width of g is at most 3 2k 1 and more importantly that there is an exponential lower bound on this relationship in particular for any k there is a graph g with treewidth equal to k where the clique width of g is at least 2 lfloor k 2 rfloor 1

some properties of infinite fuzzy relational equations with sup inf composition this paper deals with infinite fuzzy relational equations with sup inf composition first some properties of them are investigated then some necessary and sufficient conditions for the existence of attainable solutions resp unattainable solutions and partially attainable ones are given if the solution set is nonempty finally the set of attainable solutions is described

on fuzzy relational equations and the covering problem previous studies have shown that fuzzy relational equations fres based on either the max continuous archimedean t norm or the max arithmetic mean composition can be transformed into the covering problem which is an np hard problem exploiting the properties common to the continuous archimedean t norm and the arithmetic mean this study proposes a generalization of them as the u norm enabling fres that are based on the max continuous u norm composition also to be transformed into the covering problem this study also proposes a procedure for transforming the covering problem into max product fres consequently max continuous u norm fres can be solved by extending any procedure for solving either the covering problem or max product fres

bounded minimisers of double phase variational integrals bounded minimisers of the functional w mapsto int dw p a x dw q rm d x where 0 leqq a cdot in c 0 alpha and 1 p q are c 1 beta regular provided the sharp bound q leqq p alpha holds

on atomic structure of p5 free subclasses and maximum weight independent set problem graph decompositions play a crucial role in structural graph theory and in designing efficient graph algorithms among them clique separator decomposition a decomposition tree of the graph whose leaves have no clique separator so called atoms used by tarjan for solving various optimization problems recently received much attention in this note we first derive the atomic structure of two subclasses of p 5 free graphs where p 5 is a chordless path on five vertices these results enable us to provide efficient solutions for the maximum weight independent set problem in these classes of graphs

solutions and strong solutions of min product fuzzy relation inequalities with application in supply chain abstract min product fuzzy relation inequalities are introduced to describe the price requirements in the supply chain system we first investigate the resolution method and structure of the complete solution set to min product fuzzy relation inequalities motivated by the practical application we further define concept of strong solution a strong solution represents a feasible pricing scheduling enabling all the suppliers and retailers profitable in the trade activity relevant results about the strong solution are obtained based on such results a step by step algorithm is developed for finding all the strong solutions to a system of min product fuzzy relation inequalities numerical examples are given to illustrate the algorithm

restrictions of graph partition problems part i in this paper partition problems into k independent sets or cliques of bounded size k are analyzed for several classes of graphs we prove the computational complexity of both problems restricted to cographs split graphs bipartite graphs and interval graphs given general or constant k and k it is shown that the assignment problem for operations in a branching flow graph to processors each with a limit on the number of executable operations equals the first problem restricted to cographs in addition a job assignment problem given intervals for each job and k machines each executing at most k jobs equals the first problem restricted to interval graphs it is shown that both problem are np complete

a fully dynamic algorithm for the recognition of p4 sparse graphs in this paper we solve the dynamic recognition problem for the class of p 4 sparse graphs the objective is to handle edge vertex additions and deletions to recognize if each such modification yields a p 4 sparse graph and if yes to update a representation of the graph our approach relies on maintaining the modular decomposition tree of the graph which we use for solving the recognition problem we establish properties for each modification to yield a p 4 sparse graph and obtain a fully dynamic recognition algorithm which handles edge modifications in o 1 time and vertex modifications in o d time for a vertex of degree d thus our algorithm implies an optimal edges only dynamic algorithm and a new optimal incremental algorithm for p 4 sparse graphs

handsome proof nets perfect matchings and cographs we first extract the combinatorial result behind various proofs of the sequentialisation theorem for multiplicative proof nets this result is an inductive characterisation of graphs with a unique perfect matching extending these techniques we give a definition of multiplicative proof nets in which commutativity but also associativity of the multiplicative connectives is interpreted as equality this is done by representing a sequent by a cograph and axioms by a perfect matching the main advantage of this presentation is aesthetic any such graph without any further requirement is a proof structure and the correctness criterion also is a natural graph theoretical property a direct and purely graph theoretical proof of these results is available as a research report in which more details can be found c retore handsome proof nets r b graphs perfect matchings and series parallel graphs rapport de recherche rr 3652 inria march 1999 http www inria fr rrrt publications eng html

finding a potential community in networks abstract an independent 2 clique of a graph is a subset of vertices that is an independent set and such that any two vertices inside have a common neighbor outside in this paper we study the complexity of finding an independent 2 clique of maximum size in several graph classes and we compare it with the complexity of maximum independent set we prove that this problem is np hard on apex graphs apx hard on line graphs not n 1 2 approximable on bipartite graphs and not n 1 approximable on split graphs while it is polynomial time solvable on graphs of bounded degree and their complements graphs of bounded treewidth planar graphs c 3 c 6 free graphs threshold graphs interval graphs and cographs

on maximum independent set of categorical product and ultimate categorical ratios of graphs we first present polynomial algorithms to compute the independence number of the categorical product for two cographs or two splitgraphs respectively then we prove that computing the maximum independent set of the categorical product of a planar graph of maximum degree three and a k 4 is np complete the ultimate categorical independence ratio of a graph g is defined as lim k g k n k the ultimate categorical independence ratio can be computed in polynomial time for cographs splitgraphs permutation graphs interval graphs and graphs of bounded treewidth also we present an o 3 n 3 time exact exponential algorithm for the ultimate categorical independence ratio of general graphs we further present a ptas for the ultimate categorical independence ratio of planar graphs lastly we show that the ultimate categorical independent domination ratio for complete multipartite graphs is zero except when the graph is complete bipartite with color classes of equal size in which case it is 1 2

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

a simple paradigm for graph recognition application to cographs and distance hereditary graphs an easy way for graph recognition algorithms is to use a two step process first compute a characteristic feature as if the graph belongs to that class second check whether the computed characteristic feature as if the graph belongs to that class second check whether the computed separating them may yield new and much more easily understood algorithms in this paper we apply that paradigm to the cograph and distance hereditary graph recognition problems

graphs of linear clique width at most 3 abstract a graph has linear clique width at most k if it has a clique width expression using at most k labels such that every disjoint union operation has an operand which is a single vertex graph we give the first characterisation of graphs of linear clique width at most 3 and we give the first polynomial time recognition algorithm for graphs of linear clique width at most 3 in addition we present new characterisations of graphs of linear clique width at most 2 we also give a layout characterisation of graphs of bounded linear clique width a similar characterisation was independently shown by gurski and by lozin and rautenbach

note on the perfect roman domination number of graphs abstract in this note we list some propositions of the perfect roman domination number of graphs and give the characterization of graphs g with special value of the perfect roman domination furthermore r p f r p f is given at the last a linear time algorithm is shown to compute the perfect roman domination number of a cograph

on the relation between equations with max product composition and the covering problem systems of equations with max product composition are considered it is shown that solving these equations is closely related with the covering problem which belongs to the category of np hard problems it is proved that minimal solutions of equations correspond to irredundant coverings in terms of the covering problem the conditions of compatibility of equations of redundancy of equations of uniqueness of solution of uniqueness of minimal solution are determined concepts of essential non essential semi essential and super essential variables are suggested ways of simplification of a covering problem and methods of its solving are considered

exclusive graph searching vs pathwidth in graph searching a team of searchers aims at capturing an invisible fugitive moving arbitrarily fast in a graph equivalently the searchers try to clear a contaminated network the problem is to compute the minimum number of searchers required to accomplish this task several variants of graph searching have been studied mainly because of their close relationship with the pathwidth of a graph in this paper we study the complexity of the exclusive graph searching variant we show that the problem is np hard in planar graphs and it can be solved in linear time in the class of cographs we also show that monotone exclusive graph searching is np complete in split graphs where pathwidth is known to be solvable in polynomial time moreover we prove that monotone exclusive graph searching is in p in a subclass of star like graphs where pathwidth is known to be np hard

nonlinear elliptic equations with variable exponent old and new abstract in this survey paper by using variational methods we are concerned with the qualitative analysis of solutions to nonlinear elliptic problems of the type div a x u u q x 2 u in u 0 on where is a bounded or an exterior domain of r n and q is a continuous positive function the results presented in this paper extend several contributions concerning the lane emden equation and we focus on new phenomena which are due to the presence of variable exponents

locally identifying coloring of graphs with few p4s abstract a lid coloring locally identifying coloring of a graph is a proper coloring such that for any edge uv if u and v have distinct closed neighborhoods then the set of colors used on vertices of the closed neighborhoods of u and v are distinct the lid chromatic number is the minimum number of colors used in a lid coloring in this paper we prove a relation between lid coloring and a variation called strong lid coloring with this we obtain linear time algorithms to calculate the lid chromatic number for some classes of graphs with few p 4 s such as cographs p 4 sparse graphs and q q 4 graphs we also prove that the lid chromatic number is o n 1 e inapproximable in polynomial time for every e 0 unless p np

regularity for multi phase variational problems abstract we prove c 1 regularity for local minimizers of the multi phase energy w d w p a x d w q b x d w s d x under sharp assumptions relating the couples p q and p s to the holder exponents of the modulating coefficients a and b respectively

contraction and deletion blockers for perfect graphs and h free graphs abstract we study the following problem for given integers d k and graph g can we reduce some fixed graph parameter of g by at least d via at most k graph operations from some fixed set s as parameters we take the chromatic number clique number and independence number and as operations we choose edge contraction ec and vertex deletion vd we determine the complexity of this problem for s ec and s vd and for a number of subclasses of perfect graphs we use these results to determine the complexity of the problem for s ec and s vd and restricted to h free graphs

infinite fuzzy relation equations with continuous t norms this study develops a concept of infinite fuzzy relation equations with a continuous t norm it extends the work by xiong and wang q q xiong x p wang some properties of sup min fuzzy relational equations on infinite domains fuzzy sets and systems 151 2005 393 402 we describe attainable respectively unattainable solutions which are closely related to minimal solutions to the equations it is shown that a solution set comprises both attainable and unattainable solutions the study offers a characterization of these solutions under some assumptions the solution set is presented and discussed two applications are also given

some observations on maximum weight stable sets in certain p5 free graphs abstract the maximum weight stable set problem mws is the weighted version of the maximum stable set problem ms which is np hard the class of p5 free graphs i e graphs with no induced path of five vertices is the unique minimal class defined by forbidding a single connected subgraph for which the computational complexity of ms is an open question at the same time it is known that ms can be efficiently solved for p 5 f free graphs where f is any graph of five vertices different to a c5 in this paper we introduce some observations on p5 free graphs and apply them to introduce certain subclasses of such graphs for which one can efficiently solve mws that extends or improves some known results and implies together with other known results that mws can be efficiently solved for p 5 f free graphs where f is any graph of five vertices different to a c5

a theorem on permutation graphs with applications abstract in this paper a theorem that characterizes a relation between cographs and permutation graphs is introduced this theorem has an application to the recognition of a cograph by the aid of this theorem sequential and parallel algorithms to determine if a given permutation graph is a cograph can be derived

acyclic and star coloring of p4 reducible and p4 sparse graphs an acyclic coloring of a graph g is a proper vertex coloring such that g contains no bicolored cycles the more restricted notion of star coloring of g is an acyclic coloring in which each path of length 3 is not bicolored in this paper we mainly study on the acyclic and star coloring of p4 reducible and p4 sparse graphs moreover we list polynomial time algorithms for giving an optimal acyclic or star coloring of a p4 reducible or p4 sparse graph

on stability of fuzzy formal concepts over randomized one sided formal context abstract we propose a probabilistic approach to the issue of one sided fuzzy formal concepts stability the modified rice siff algorithm represents a crisp index how to select the relevant concepts from the set of all one sided fuzzy formal concepts we suggest to explore the formal concepts stability affected by the random fluctuation of values in a formal context we describe the algorithm and study the properties of the concept stability using random variables with the gaussian normal distribution in combination with the modified rice siff algorithm the gaussian probabilistic index improves the analysis of the most relevant one sided formal concepts from the original one sided formal context the connections to recent works in the related directions are presented