229 research outputs found

    Coloring Artemis graphs

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    We consider the class A of graphs that contain no odd hole, no antihole, and no ``prism'' (a graph consisting of two disjoint triangles with three disjoint paths between them). We show that the coloring algorithm found by the second and fourth author can be implemented in time O(n^2m) for any graph in A with n vertices and m edges, thereby improving on the complexity proposed in the original paper

    Precoloring co-Meyniel graphs

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    The pre-coloring extension problem consists, given a graph GG and a subset of nodes to which some colors are already assigned, in finding a coloring of GG with the minimum number of colors which respects the pre-coloring assignment. This can be reduced to the usual coloring problem on a certain contracted graph. We prove that pre-coloring extension is polynomial for complements of Meyniel graphs. We answer a question of Hujter and Tuza by showing that ``PrExt perfect'' graphs are exactly the co-Meyniel graphs, which also generalizes results of Hujter and Tuza and of Hertz. Moreover we show that, given a co-Meyniel graph, the corresponding contracted graph belongs to a restricted class of perfect graphs (``co-Artemis'' graphs, which are ``co-perfectly contractile'' graphs), whose perfectness is easier to establish than the strong perfect graph theorem. However, the polynomiality of our algorithm still depends on the ellipsoid method for coloring perfect graphs

    Perfect Graphs

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    This chapter is a survey on perfect graphs with an algorithmic flavor. Our emphasis is on important classes of perfect graphs for which there are fast and efficient recognition and optimization algorithms. The classes of graphs we discuss in this chapter are chordal, comparability, interval, perfectly orderable, weakly chordal, perfectly contractile, and chi-bound graphs. For each of these classes, when appropriate, we discuss the complexity of the recognition algorithm and algorithms for finding a minimum coloring, and a largest clique in the graph and its complement

    The stable set polytope and some operations on graphs

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    AbstractWe study some operations on graphs in relation to the stable set polytope, for instance, identification of two nodes, linking a pair of nodes by an edge and composition of graphs by subgraph identification. We show that, with appropriate conditions, the descriptions of the stable set polytopes associated with the resulting graphs can be derived from those related to the initial graphs by adding eventual clique inequalities. Thus, perfection and h-perfection of graphs are preserved

    Ki-covers I: Complexity and polytopes

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    AbstractA Ki in a graph is a complete subgraph of size i. A Ki-cover of a graph G(V, E is a set C of Ki − 1's of G such that every Ki in G contains at least one Ki − 1 in C. Thus a K2-cover is a vertex cover. The problem of determining whether a graph has a Ki-cover (i ⩾ 2) of cardinality ⩽k is shown to be NP-complete for graphs in general. For chordal graphs with fixed maximum clique size, the problem is polynomial; however, it is NP-complete for arbitrary chordal graphs when i ⩾ 3. The NP-completeness results motivate the examination of some facets of the corresponding polytope. In particular we show that various induced subgraphs of G define facets of the Ki-cover polytope. Further results of this type are also produced for the K3-cover polytope. We conclude by describing polynomial algorithms for solving the separation problem for some classes of facets of the Ki-cover polytope

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