On the complexity of minimum cardinality maximal uniquely restricted matching in graphs

Abstract

For a graph G =(V,E), a set M ⊆ E is called a matching in G if no two edges in M share a common vertex. A matching M in G is called an uniquely restricted matching in G if there is no other matching of the same cardinality in the graph induced on the vertices saturated by M. An uniquely restricted matching M is called maximal if M is not properly contained in any uniquely restricted matching of G. The minimum maximal uniquely restricted matching (Min-UR-Matching) problem is the problem of finding a minimum cardinality maximal uniquely restricted matching. In this paper, we initiate the study of theMin-UR-Matching problem. We prove that the decision version of the Min-UR-Matching problem is NP-complete for general graphs. In particular, this answers an open question posed by Hedetniemi [AKCE J. Graphs. Combin. 3(1)(2006) 1–37] regarding the complexity of the Min-UR-Matching problem. We also prove that this problem remains NP-complete for bipartite graphs with maximum degree 7. Next, we show that the Min-UR-Matching for bipartite graphs cannot be approximated within a factor of n1-ϵ for any positive constant ϵ >0 unless P = NP. Finally, we prove that the Min-UR-Matching problem is linear-time solvable for chain graphs, a subclass of bipartite graphs.