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)>log38×k≈1.893k. Note that mmw(G)≤tw(G)+1≤3mmw(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)