Maximum Induced Matching Algorithms via Vertex Ordering Characterizations

Abstract

We study the maximum induced matching problem on a graph G. Induced matchings correspond to independent sets in \(L^2(G)\), the square of the line graph of G. The problem is NP-complete on bipartite graphs. In this work, we show that for a number of graph families characterized by vertex orderings, almost all forbidden patterns on three vertices are preserved when taking the square of the line graph. These orderings can be computed in linear time in the size of the input graph. In particular, given \(\mathcal {G}\) a graph class, and a graph \(G=(V,E) \in \mathcal {G}\) with a corresponding vertex ordering \(\sigma \) of V, one can produce (in linear time in the size of G) an ordering on the vertices of \(L^2(G)\), that shows that \(L^2(G) \in \mathcal {G}\) without computing the line graph or the square of the line graph of G. These results generalize and unify previous ones on showing closure under \(L^2(\cdot )\) for various graph families. Furthermore, these orderings on \(L^2(G)\) can be exploited algorithmically to compute a maximum induced matching on G faster. We illustrate this latter fact in the second half of the paper where we focus on cocomparability graphs, a large graph class that includes interval, permutation, trapezoid graphs, and co-graphs, and we present the first \(\mathcal {O}(mn)\) time algorithm to compute a maximum weighted induced matching on cocomparability graphs; an improvement from the best known \(\mathcal {O}(n^4)\) time algorithm for the unweighted case.