On the P-vertex problem in Bipartite Graphs

In a recent work, Sharma and Panda~\cite{sharma} showed that every bipartite graph with a perfect matching has property (P). In this paper, we investigate the converse direction, i.e., when property (P) forces the existence of a perfect matching in bipartite graphs. We show that such graphs are balanced and establish that property (P) is equivalent to the existence of a perfect matching for several families of bipartite graphs.