Elsevier

Discrete Applied Mathematics

Volume 248, 30 October 2018, Pages 114-124
Discrete Applied Mathematics

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

https://doi.org/10.1016/j.dam.2017.09.019Get rights and content
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Abstract

A graph of treewidth k has a representation by subtrees of a ternary tree, with subtrees of adjacent vertices sharing a tree node, and any tree node sharing at most k+1 subtrees. Likewise for branchwidth, but with a shift to the edges of the tree rather than the nodes. In this paper we show that the mm-width of a graph – maximum matching width – combines aspects of both these representations, targeting tree nodes for adjacency and tree edges for the parameter value. The proof of this new characterization of mm-width is based on a definition of canonical minimum vertex covers of bipartite graphs. We show that these behave in a monotone way along branch decompositions over the vertex set of a graph.

We use these representations to compare mm-width with treewidth and branchwidth, and also to give another new characterization of mm-width, by subgraphs of chordal graphs. We prove that given a graph G and a branch decomposition of maximum matching width k we can solve the Minimum Dominating Set Problem in time O(8k), thereby beating O(3tw(G)) whenever tw(G)>log38×k1.893k. Note that mmw(G)tw(G)+13mmw(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 Minimum Dominating Set whenever tw(G)>1.549×mmw(G).

Keywords

Treewidth
Branchwidth
Maximum matching width
mm-width
Minimum Dominating Set Problem

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