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On Linear and Circular Structure of (claw, net)Free Graphs
, 2003
"... We prove that every (claw, net)free graph contains an induced doubly dominating cycle or a dominating pair. Moreover, using LexBFS we present alS[SE timealen##ES which, for a given (claw, net)free graph, finds either a dominating pair or an induceddoubl dominatingcycln We show aln how one can uses ..."
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We prove that every (claw, net)free graph contains an induced doubly dominating cycle or a dominating pair. Moreover, using LexBFS we present alS[SE timealen##ES which, for a given (claw, net)free graph, finds either a dominating pair or an induceddoubl dominatingcycln We show aln how one can usestructural properties of (claw, net)free graphs tosolI efficiently the domination, independent domination, and independent set problems on these graphs.
On the algorithmic complexity of twelve covering and independence parameters of graphs
 Discrete Applied Mathematics
, 1999
"... The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning ..."
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The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned twelve covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NPcompleteness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs.
LexBFSordering in Asteroidal Triplefree Graphs
 SpringerVerlag Lecture Notes in Computer Science 1741
, 1999
"... . In this paper, we study the metric property of LexBFSordering on ATfree graphs. Based on a 2sweep LexBFS algorithm, we show that every ATfree graph admits a vertex ordering, called the strong 2cocomparability ordering, that for any three vertices u OE v OE w in the ordering, if d(u; w) 2 ..."
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. In this paper, we study the metric property of LexBFSordering on ATfree graphs. Based on a 2sweep LexBFS algorithm, we show that every ATfree graph admits a vertex ordering, called the strong 2cocomparability ordering, that for any three vertices u OE v OE w in the ordering, if d(u; w) 2 then d(u; v) = 1 or d(v; w) 2. As an application of this ordering, we provide a simple linear time recognition algorithm for bipartite permutation graphs, which form a subclass of ATfree graphs. 1 Introduction In past years, specific orderings of vertices characterizing certain graph classes are studied by many researchers. Usually, these ordering can be described from a metric point of view and the metric associated with a connected graph is, of course, the distance function d, giving the length of a shortest path between two vertices. One of the first results is due to Rose [19] for recognizing chordal graphs. A graph is chordal if every cycle of length at least four has a chord. A p...
A branchandreduce algorithm for finding a minimum independent dominating set in graphs
 In International Workshop on GraphTheoretic Concepts in Computer Science (WG
, 2006
"... Abstract. A dominating set D of a graph G = (V, E) is a subset of vertices such that every vertex in V \ D has at least one neighbour in D. Moreover if D is an independent set, i.e. no vertices in D are pairwise adjacent, then D is said to be an independent dominating set. Finding a minimum independ ..."
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Abstract. A dominating set D of a graph G = (V, E) is a subset of vertices such that every vertex in V \ D has at least one neighbour in D. Moreover if D is an independent set, i.e. no vertices in D are pairwise adjacent, then D is said to be an independent dominating set. Finding a minimum independent dominating set in a graph is an NPhard problem. We give an algorithm computing a minimum independent dominating set of a graph on n vertices in time O(1.3575 n). Furthermore, we show that Ω(1.3247 n) is a lower bound on the worstcase running time of this algorithm. 1
Kayles and Nimbers
, 2000
"... Kayles is a combinatorial game on graphs. Two players select alternatingly a vertex from a given graph G  a chosen vertex may not be adjacent or equal to an already chosen vertex. The last player that can select a vertex wins the game. The problem to determine which player has a winning strategy is ..."
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Kayles is a combinatorial game on graphs. Two players select alternatingly a vertex from a given graph G  a chosen vertex may not be adjacent or equal to an already chosen vertex. The last player that can select a vertex wins the game. The problem to determine which player has a winning strategy is known to be PSPACEcomplete. Because of certain characteristics of the Kayles game, it can be analyzed with SpragueGrundy theory. In this way, we can show that the problem is polynomial time solvable for graphs with a bounded asteroidal number. It is shown that the problem can be solved in O(n^3) time on cocomparability graphs and circular arc graphs, and in O(n 1 1/ log 3) = O(n^1.631) time on cographs.
A penaltyevaporation heuristic in a decomposition method for the maximum clique problem
 In Optimization Days
, 2003
"... In this paper, we present a heuristic method to solve the maximum clique problem, based on the concepts of penalty and evaporation. At each iteration, some vertex i is inserted into the current solution (always a clique) and the vertices that are not adjacent to vertex i are removed from the solutio ..."
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In this paper, we present a heuristic method to solve the maximum clique problem, based on the concepts of penalty and evaporation. At each iteration, some vertex i is inserted into the current solution (always a clique) and the vertices that are not adjacent to vertex i are removed from the solution. The removed vertices are then penalized in order to reduce their potential of being selected to be inserted in the solution again during the next iterations. This penalty is gradually evaporating to allow vertices to become interesting subsequently. This penaltyevaporation heuristic method is embedded in a decomposition algorithm that restricts the search for a maximum clique to subgraphs, but performs an aggressive exploration of the feasible domain. Numerical results indicate that the penaltyevaporation heuristic method alone is effective and reliable, but the gain in quality obtained when embedding it in the decomposition algorithm is worthy of the additional computing time required. 1 1
The Dichotomy of List Homomorphisms for Digraphs
"... The DichotomyConjecture for Constraint Satisfaction Problems has been verified for conservative problems (or, equivalently, for list homomorphism problems) by Andrei Bulatov. An earlier case of this dichotomy, for list homomorphisms to undirected graphs, came with an elegant structural distinction b ..."
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The DichotomyConjecture for Constraint Satisfaction Problems has been verified for conservative problems (or, equivalently, for list homomorphism problems) by Andrei Bulatov. An earlier case of this dichotomy, for list homomorphisms to undirected graphs, came with an elegant structural distinction between the tractable and intractable cases. Such structural characterization is absent in Bulatov’s classification, and Bulatov asked whether one can be found. We provide an answer in the case of digraphs. In the process we give forbidden structure characterizations of the existence of certain polymorphisms relevant in Bulatov’s dichotomy classification. The key concept we introduce is that of a digraph asteroidal triple (DAT). The dichotomy then takes the following form. If a digraph H has a DAT, then the list homomorphism
Feedback vertex set on ATfree graphs
, 2007
"... We present a polynomial time algorithm to compute a minimum (weight) feedback vertex set for ATfree graphs, and extending this approach we obtain a polynomial time algorithm for graphs of bounded asteroidal number. ..."
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We present a polynomial time algorithm to compute a minimum (weight) feedback vertex set for ATfree graphs, and extending this approach we obtain a polynomial time algorithm for graphs of bounded asteroidal number.
Classes of Perfect Graphs
, 2003
"... Abstract. The Strong Perfect Graph Conjecture, suggested by Claude Berge in 1960, had a major impact on the development of graph theory over the last forty years. It has led to the definitions and study of many new classes of graphs for which the Strong Perfect Graph Conjecture has been verified. Po ..."
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Abstract. The Strong Perfect Graph Conjecture, suggested by Claude Berge in 1960, had a major impact on the development of graph theory over the last forty years. It has led to the definitions and study of many new classes of graphs for which the Strong Perfect Graph Conjecture has been verified. Powerful concepts and methods have been developed to prove the Strong Perfect Graph Conjecture for these special cases. In this paper we survey 120 of these classes, list their fundamental algorithmic properties and present all known relations between them. 1