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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 showdoi:10.1016/j.dam.2017.09.019 fatcat:heppqi43ffct7nrqbmxua7ulka