23,961 research outputs found

    On the algorithmic complexity of twelve covering and independence parameters of graphs

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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 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness 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

    Induced subgraphs of graphs with large chromatic number. XII. Distant stars

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    The Gyarfas-Sumner conjecture asserts that if H is a tree then every graph with bounded clique number and very large chromatic number contains H as an induced subgraph. This is still open, although it has been proved for a few simple families of trees, including trees of radius two, some special trees of radius three, and subdivided stars. These trees all have the property that their vertices of degree more than two are clustered quite closely together. In this paper, we prove the conjecture for two families of trees which do not have this restriction. As special cases, these families contain all double-ended brooms and two-legged caterpillars

    Reconfiguration of Dominating Sets

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    We explore a reconfiguration version of the dominating set problem, where a dominating set in a graph GG is a set SS of vertices such that each vertex is either in SS or has a neighbour in SS. In a reconfiguration problem, the goal is to determine whether there exists a sequence of feasible solutions connecting given feasible solutions ss and tt such that each pair of consecutive solutions is adjacent according to a specified adjacency relation. Two dominating sets are adjacent if one can be formed from the other by the addition or deletion of a single vertex. For various values of kk, we consider properties of Dk(G)D_k(G), the graph consisting of a vertex for each dominating set of size at most kk and edges specified by the adjacency relation. Addressing an open question posed by Haas and Seyffarth, we demonstrate that DΓ(G)+1(G)D_{\Gamma(G)+1}(G) is not necessarily connected, for Γ(G)\Gamma(G) the maximum cardinality of a minimal dominating set in GG. The result holds even when graphs are constrained to be planar, of bounded tree-width, or bb-partite for b≥3b \ge 3. Moreover, we construct an infinite family of graphs such that Dγ(G)+1(G)D_{\gamma(G)+1}(G) has exponential diameter, for γ(G)\gamma(G) the minimum size of a dominating set. On the positive side, we show that Dn−m(G)D_{n-m}(G) is connected and of linear diameter for any graph GG on nn vertices having at least m+1m+1 independent edges.Comment: 12 pages, 4 figure

    Isomorphism of graph classes related to the circular-ones property

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    We give a linear-time algorithm that checks for isomorphism between two 0-1 matrices that obey the circular-ones property. This algorithm leads to linear-time isomorphism algorithms for related graph classes, including Helly circular-arc graphs, \Gamma-circular-arc graphs, proper circular-arc graphs and convex-round graphs.Comment: 25 pages, 9 figure
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