Computing unique maximum matchings in $O(m)$ time for König–Egerváry graphs and unicyclic graphs

Abstract

Let $\alpha \left( G\right)$ denote the maximum size of an independent set of vertices and $\mu \left( G\right)$ be the cardinality of a maximum matching in a graph $G$. A matching saturating all the vertices is a perfect matching. If $\alpha \left( G\right) +\mu \left( G\right) =\left| V(G)\right|$, then $G$ is called a König–Egerváry graph. A graph is unicyclic if it is connected and has a unique cycle. It is known that a maximum matching can be found in $O(m\cdot \sqrt{n})$ time for a graph with $n$ vertices and $m$ edges. Bartha (Proceedings of the 8th joint conference on mathematics and computer science, Komárno, Slovakia, 2010) 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 (Karp and Sipser in Proceedings of the 22nd annual IEEE symposium on foundations of computer science, 364–375, 1981) , 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.