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 \( URM \) $\mathit{URM}$ -extendable) if every uniquely restricted matching is included in a perfect matching. A \( URM \) $\mathit{URM}$ -extendable graph G is minimal if \(G-e\) $G-e$ is not \( URM \) $\mathit{URM}$ -extendable for any edge e. In this paper, we find all \( URM \) $\mathit{URM}$ -extendable cubic graphs. For any integer \(r\ge 1\) $r\ge 1$ , we construct some \((2r+1)\) $(2r+1)$ -regular \( URM \) $\mathit{URM}$ -extendable graphs and \((4r+2)\) $(4r+2)$ -regular \( URM \) $\mathit{URM}$ -extendable graphs. We show that \(T\otimes K_2\) $T\otimes K_2$ is a minimal \( URM \) $\mathit{URM}$ -extendable graph for any tree T. Finally, we show that the minimum size of \( URM \) $\mathit{URM}$ -extendable graph G on 2n vertices is \(4n-4\) $4n-4$ and characterize the extreme graph.