Computing Unique Maximum Matchings in O(m) time for Konig-Egervary Graphs and Unicyclic Graphs

Let α (G) denote the maximum size of an independent set of vertices and μ (G) be the cardinality of a maximum matching in a graph G. A matching saturating all the vertices is a perfect matching. If α (G) + μ (G) = |V (G)|, then G is called a König-Egerváry graph. A graph is unicyclic if it has a unique cycle. It is known that a maximum matching can be found in O(m •√n) time for a graph with n vertices and m edges. Bartha [1] conjectured that a unique perfect matching, if it exists, can be found in O(m) time. In this paper we validate this conjecture for König-Egerváry graphs and unicylic graphs. We propose a variation of Karp-Sipser leaf-removal algorithm [11], which ends with an empty graph if and only if the original graph is a König-Egerváry graph with a unique perfect matching (obtained as an output as well). We also show that a unicyclic non-bipartite graph G may have at most one perfect matching, and this is the case where G is a König-Egerváry graph.