On the complexity of minimum maximal uniquely restricted matching

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. A matching M in a graph G is called a uniquely restricted matching if, G[V(M)], the subgraph of G induced by the set of M-saturated vertices of G contains exactly one perfect matching. A uniquely restricted matching M is maximal if M is not properly contained in any uniquely restricted matching of G. Given a graph G, the MIN-MAX-UR MATCHING problem asks to find a maximal uniquely restricted matching in G of minimum cardinality and DECIDE-MIN-MAX-UR MATCHING problem, the decision version of this problem takes a graph G and an integer k and asks whether G admits a maximal uniquely restricted matching of cardinality at most k. It is known that the DECIDE-MIN-MAX-UR MATCHING problem is NP-complete. In this paper, we strengthen this result by proving that the DECIDE-MIN-MAX-UR MATCHING problem remains NP-complete for chordal bipartite graphs, star-convex bipartite graphs, chordal graphs, and doubly chordal graphs. On the positive side, we prove that the MIN-MAX-UR MATCHING problem is polynomial time solvable for bipartite distance-hereditary graphs and linear time solvable for bipartite permutation graphs, proper interval graphs, and threshold graphs. Finally, we prove that the MIN-MAX-UR MATCHING problem is APX-complete for graphs with maximum degree 4.