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

Vadim E. Levit, Eugen Mandrescu

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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 is connected and 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 (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.

Original languageEnglish
Pages (from-to)267-277
Number of pages11
JournalJournal of Combinatorial Optimization
Issue number1
StatePublished - 1 Jul 2016
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2015, Springer Science+Business Media New York.


  • Core
  • Karp–Sipser leaf-removal algorithm
  • Konig–Egervary graph
  • Unicyclic graph
  • Unique perfect matching


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