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

Abstract

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.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.

On the König deficiency of zero-reducible graphs

“…As mentioned in Sect. 1, both components have been studied earlier separately in Levit and Mandrescu (2016) and Krész (2006, 2009), respectively, and it has been observed that they preserve the number of perfect matchings in graphs. Moreover, leaf reduction (elimination) preserves the König deficiency as well.…”

“…Leaf reduction by itself can be carried out by a straightforward linear-time (O(m)) algorithm. For a concrete implementation, see Levit and Mandrescu (2016). Thus, zero-leaf-reducibility of graphs is decidable in linear time.…”

“…We start out by providing a simple alternative proof of the characterization result obtained by Levit and Mandrescu (2016) on UPM graphs having the König property. In our language, the characterization says that these graphs are exactly the zero-leafreducible ones.…”

“…In a recent paper, Levit and Mandrescu (2016) have given an interesting characterization of graphs having a unique perfect matching and satisfying the König property The second author acknowledges the European Commission for funding the InnoRenew CoE project (Grant Agreement #739574) under the Horizon2020 Widespread-Teaming program and the Republic of Slovenia (Investment funding of the Republic of Slovenia and the European Union of the European regional Development Fund). Miklós Krész is also grateful for the support of the EU-funded Hungarian grant EFOP-3.6.2-16-2017-00015 and the Slovenian ARRS grant J1-9110.…”

“…We also investigate the possibility whether, on the analogy of the main result of Levit and Mandrescu (2016), one could come up with a characterization of zeroreducible graphs in terms of the König property. Unfortunately it turns out that such a characterization is not likely to exist.…”