Diverse Pairs of Matchings

Abstract

AbstractWe initiate the study of the Diverse Pair of (Maximum/ Perfect) Matchings problems which given a graph G and an integer k, ask whether G has two (maximum/perfect) matchings whose symmetric difference is at least k. Diverse Pair of Matchings (asking for two not necessarily maximum or perfect matchings) is $$\textsf{NP}$$ NP -complete on general graphs if k is part of the input, and we consider two restricted variants. First, we show that on bipartite graphs, the problem is polynomial-time solvable, and second we show that Diverse Pair of Maximum Matchings is $$\textsf{FPT}$$ FPT parameterized by k. We round off the work by showing that Diverse Pair of Matchings has a kernel on $${\mathcal {O}}(k^2)$$ O ( k 2 ) vertices.