Matchability and k-maximal matchings

We present a collection of new structural, algorithmic, and complexity results for two types of matching problems. The first problem involves the computation of k-maximal matchings, where a matching is k-maximal if it admits no augmenting path with ≤ 2k vertices. The second involves finding a maximal set of vertices that is matchable — comprising one side of the edges in some matching. Among our results, we prove that the minimum cardinality β2 of a 2-maximal matching is at most the minimum cardinality μ of a maximal matchable set, with equality attained for triangle-free graphs. We show that the parameters β2 and μ are NP-hard to compute in bipartite and chordal graphs, but can be computed in linear time on a tree. Finally, we also give a simple linear-time algorithm for finding a 3-maximal matching, a consequence of which is a simple linear-time 3/4-approximation algorithm for the maximum-cardinality matching problem in a general graph.