Diverse Pairs of Matchings

We 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 \(\textsf{NP}\) $\mathsf{NP}$ -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 \(\textsf{FPT}\) $\mathsf{FPT}$ parameterized by k. We round off the work by showing that Diverse Pair of Matchings has a kernel on \({\mathcal {O}}(k^2)\) $\mathcal{O}(k^2)$ vertices.