On distance-3 matchings and induced matchings

• Authors: Andreas Brandstädt;Raffaele Mosca
• Venue: Discrete Applied Mathematics
• Year: 2011

For a finite undirected graph G=(V,E) and positive integer k=1, an edge set M@?E is a distance-k matching if the pairwise distance of edges in M is at least k in G. For k=1, this gives the usual notion of matching in graphs, and for general k=1, distance-k matchings were called k-separated matchings by Stockmeyer and Vazirani. The special case k=2 has been studied under the names induced matching (i.e., a matching which forms an induced subgraph in G) by Cameron and strong matching by Golumbic and Laskar in various papers. Finding a maximum induced matching is NP-complete even on very restricted bipartite graphs and on claw-free graphs but it can be done efficiently on various classes of graphs such as chordal graphs, based on the fact that an induced matching in G corresponds to an independent vertex set in the square L(G)^2 of the line graph L(G) of G which, by a result of Cameron, is chordal for any chordal graph G. We show that, unlike for k=2, for a chordal graph G, L(G)^3 is not necessarily chordal, and finding a maximum distance-3 matching, and more generally, finding a maximum distance-(2k+1) matching for k=1, remains NP-complete on chordal graphs. For strongly chordal graphs and interval graphs, however, the maximum distance-k matching problem can be solved in polynomial time for every k=1. Moreover, we obtain various new results for maximum induced matchings on subclasses of claw-free graphs.