Acyclic matching in some subclasses of graphs

Abstract

A subset M⊆E of edges of a graph G=(V,E) is called a matching if no two edges of M share a common vertex. Given a matching M in G, a vertex v∈V is called M-saturated if there exists an edge e∈M incident with v. A matching M of a graph G is called an acyclic matching if, G[V(M)], the subgraph of G induced by the M-saturated vertices of G is an acyclic graph. Given a graph G, the Acyclic Matching problem asks to find an acyclic matching of maximum cardinality in G. The Decide-Acyclic Matching problem takes a graph G and an integer k and asks whether G has an acyclic matching of cardinality at least k. The Decide-Acyclic Matching problem is known to be NP-complete for general graphs as well as for bipartite graphs. In this paper, we strengthen this result by showing that the Decide-Acyclic Matching problem remains NP-complete for comb-convex bipartite graphs, star-convex bipartite graphs, and dually chordal graphs. On the positive side, we show that the Acyclic Matching problem is linear time solvable for split graphs, block graphs, and proper interval graphs. We show that the Acyclic Matching problem is hard to approximate within a factor of n1−ϵ for any ϵ>0 unless P=NP. Also, we show that the Acyclic Matching problem is APX-complete for (2k+1)-regular graphs for every fixed integer k≥3.