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We give alternative definitions for maximum matching width, e.g. a graph $G$has $\operatorname{mmw}(G) \leq k$ if and only if it is a subgraph of a chordalgraph $H$ and for every maximal clique $X$ of $H$ there exists $A,B,C \subseteqX$ with $A \cup B \cup C=X$ and $|A|,|B|,|C| \leq k$ 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 ofsubtrees, where treewidth focuses on nodes and branchwidth focuses on edges. Weshow that mm-width combines both aspects, focusing on nodes and on edges. Basedon 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$O^*(3^{\operatorname{tw}(G)})$ whenever $\operatorname{tw}(G) > \log_3{8}\times k \approx 1.893 k$. Note that $\operatorname{mmw}(G) \leq\operatorname{tw}(G)+1 \leq 3 \operatorname{mmw}(G)$ and these inequalities aretight. Given only the graph $G$ and using the best known algorithms to finddecompositions, maximum matching width will be better for solving DominatingSet whenever $\operatorname{tw}(G) > 1.549 \times \operatorname{mmw}(G)$.

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