1,029 research outputs found

    Rank-width and Tree-width of H-minor-free Graphs

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    We prove that for any fixed r>=2, the tree-width of graphs not containing K_r as a topological minor (resp. as a subgraph) is bounded by a linear (resp. polynomial) function of their rank-width. We also present refinements of our bounds for other graph classes such as K_r-minor free graphs and graphs of bounded genus.Comment: 17 page

    Maximum matching width: new characterizations and a fast algorithm for dominating set

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    We give alternative definitions for maximum matching width, e.g. a graph GG has mmw(G)k\operatorname{mmw}(G) \leq k if and only if it is a subgraph of a chordal graph HH and for every maximal clique XX of HH there exists A,B,CXA,B,C \subseteq X with ABC=XA \cup B \cup C=X and A,B,Ck|A|,|B|,|C| \leq k such that any subset of XX that is a minimal separator of HH is a subset of either A,BA, B or CC. Treewidth and branchwidth have alternative definitions through intersections of subtrees, where treewidth focuses on nodes and branchwidth focuses on edges. We show that mm-width combines both aspects, focusing on nodes and on edges. Based on this we prove that given a graph GG and a branch decomposition of mm-width kk we can solve Dominating Set in time O(8k)O^*({8^k}), thereby beating O(3tw(G))O^*(3^{\operatorname{tw}(G)}) whenever tw(G)>log38×k1.893k\operatorname{tw}(G) > \log_3{8} \times k \approx 1.893 k. Note that mmw(G)tw(G)+13mmw(G)\operatorname{mmw}(G) \leq \operatorname{tw}(G)+1 \leq 3 \operatorname{mmw}(G) and these inequalities are tight. Given only the graph GG and using the best known algorithms to find decompositions, maximum matching width will be better for solving Dominating Set whenever tw(G)>1.549×mmw(G)\operatorname{tw}(G) > 1.549 \times \operatorname{mmw}(G)
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