On Distance-3 Matchings and Induced Matchings

For a finite undirected graph G = (V, E) and positive integer k >= 1, an edge set M subset of E is a distance-k matching if the mutual 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 but for k = 2 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 remains NP-complete on chordal graphs. For strongly chordal graphs and interval graphs, however, the maximum distance-3 matching problem can be solved in polynomial time. Moreover, we obtain various new results for induced matchings.