Parameterized complexity of perfectly matched sets

For an undirected graph G, a pair of vertex disjoint subsets (A,B) is a pair of perfectly matched sets if each vertex in A (resp. B) has exactly one neighbor in B (resp. A). In the above, the size of the pair is |A| (=|B|). Given a graph G and a positive integer k, the PERFECTLY MATCHED SETS problem asks whether there exists a pair of perfectly matched sets of size at least k in G. This problem is known to be NP-hard on planar graphs and W[1]-hard on general graphs, when parameterized by k. However, little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we study the problem parameterized by k, and design FPT algorithms for: i) apex-minor-free graphs running in time 2^O(k)⋅n^O(1), and ii) K_b,b-free graphs. We obtain a linear kernel for planar graphs and k^O(d)-sized kernel for d-degenerate graphs. It is known that the problem is W[1]-hard on chordal graphs, in fact on split graphs, parameterized by k. We complement this hardness result by designing a polynomial-time algorithm for interval graphs.