Uniquely Restricted Matching Extendable Graphs

A matching M of a graph G is called uniquely restricted if M is a unique perfect matching of the subgraph induced by M-saturated vertex set. A connected graph is called uniquely restricted matching extendable (abbreviated as URM -extendable) if every uniquely restricted matching is included in a perfect matching. A URM -extendable graph G is minimal if G-e is not URM -extendable for any edge e. In this paper, we find all URM -extendable cubic graphs. For any integer r≥ 1 , we construct some (2r+1) -regular URM -extendable graphs and (4r+2) -regular URM -extendable graphs. We show that T⊗ K_2 is a minimal URM -extendable graph for any tree T. Finally, we show that the minimum size of URM -extendable graph G on 2n vertices is 4n-4 and characterize the extreme graph.