Maximum induced matching problem on hhd-free graphs

An induced matching in a graph is a set of edges such that no two edges in the set are joined by any third edge of the graph. An induced matching is maximum (MIM) if the number of edges in it is the largest among all possible induced matchings. It is known that finding the size of a MIM in a graph is NP-hard even if the graph is bipartite. It is also known that the size of a MIM in a chordal graph or in a weakly chordal graph can be computed in polynomial time. Specifically, the size of a MIM can be computed in linear time for a chordal graph and in O(m) time for a weakly chordal graph. This work demonstrates some algorithms for the maximum induced matching problem with better complexity than O(m) for some subclasses of weakly chordal graphs. In particular, we show that the maximum induced matching problem can be solved on hhd-free graphs (graphs without the complement of a P5 (house), induced cycle on 5 or more vertices (hole), or the bipartite graph obtained from a C6 by adding a single chord (domino)) in O(m ) time; hhd-free graphs generalize chordal graphs. Then, we consider a technique used by Brandsädt and Hoàng to solve the problem on chordal graphs. Extending this we show that for two subclasses of hhd-free graphs, the problem can be solved in better time than O(m). The first is the class of bipartite graphs that are hhd-free for which we obtain an O(mn) time algorithm. The other class of graphs we consider, for which we obtain a linear time algorithm, is more general than chordal graphs.