Maximum Matching Width: New Characterizations and a Fast Algorithm for Dominating Set

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 ⊆ X with A∪B∪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∗(8k), thereby beating O∗(3tw(G)) whenever tw(G) > log3 8×k ≈ 1.893k. 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). 1998 ACM Subject Classification G.2.2 Graph Theory