Maximum Induced Matching Algorithms via Vertex Ordering Characterizations

TL;DR

This paper introduces linear-time methods to analyze and compute maximum induced matchings in specific graph classes by leveraging vertex orderings, leading to faster algorithms especially for cocomparability graphs.

Contribution

It presents a novel approach to preserve forbidden vertex orderings through graph transformations, enabling efficient maximum induced matching algorithms.

Findings

Orderings can be computed in linear time for various graph classes.

L2(g) remains within the same graph class for many forbidden ordering families.

First O(mn) algorithm for maximum weighted induced matching in cocomparability graphs.

Abstract

We study the maximum induced matching problem on a graph g. Induced matchings correspond to independent sets in L2(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 with forbidden 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 a graph class G characterized by a vertex ordering, and a graph g = (V, E) in 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 L2(g), that shows that L2(g) in G - for a number of graph classes 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 L2(.) for various graph families. Furthermore, these orderings on L2(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 O(mn) time algorithm to compute a maximum weighted induced matching on cocomparability graphs; an improvement from the best known O(n4) time algorithm for the unweighted case.