On Unicyclic Graphs with Uniquely Restricted Maximum Matchings

A graph is called unicyclic if it owns only one cycle. A matching M is called uniquely restricted in a graph G if it is the unique perfect matching of the subgraph induced by the vertices that M saturates. Clearly, μ_r(G) ≤ μ(G), where μ_r(G) denotes the size of a maximum uniquely restricted matching, while μ(G) equals the matching number of G. In this paper we study unicyclic bipartite graphs enjoying μ_r(G) = μ(G). In particular, we characterize unicyclic bipartite graphs having only uniquely restricted maximum matchings. Finally, we present some polynomial time algorithms recognizing unicyclic bipartite graphs with (only) uniquely restricted maximum matchings.