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

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Conference object###### Peer reviewed

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##### Date

2015

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We give alternative definitions for maximum matching width, e.g., a graph G has mmw(G) <= k if and only if it is a subgraph of a chordal graph H and for every maximal clique X of H there exists A,B,C \subseteq X with A \cup B \cup C=X and |A|,|B|,|C| <= k such that any subset of X that is a minimal separator of H is a subset of either A, B or C. 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 G and a branch decomposition of mm-width k we can solve Dominating Set in time O^*(8^k), thereby beating O^*(3^{tw(G)}) whenever tw(G) > log_3(8) * k ~ 1.893 k. Note that mmw(G) <= tw(G)+1 <= 3 mmw(G) and these inequalities are tight. Given only the graph G 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).

##### Citation

Leibniz International Proceedings in Informatics 2015, 43:212-223##### Publisher

Dagstuhl Publishing##### Collections

Copyright Jisu Jeong, Sigve Hortemo Sæther, and Jan Arne Telle