  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 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 $|A|,|B|,|C| \leq k$ 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 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
$O^*(3^{\operatorname{tw}(G)})$ 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 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 $\operatorname{tw}(G) > 1.549 \times \operatorname{mmw}(G)$.
