Perfectly matched sets in graphs: Parameterized and exact computation

In an undirected graph G=(V,E), we say that (A,B) is a pair of perfectly matched sets if A and B are disjoint subsets of V and every vertex in A (resp. B) has exactly one neighbor in B (resp. A). The size of a pair of perfectly matched sets (A,B) is |A|=|B|. The PERFECTLY MATCHED SETS problem is to decide whether a given graph G has a pair of perfectly matched sets of size k. We show that PMS is W[1]-hard when parameterized by solution size k even when restricted to split graphs and bipartite graphs. We observe that PMS is FPT when parameterized by clique-width, and give FPT algorithms with respect to the parameters distance to cluster, distance to co-cluster and treewidth. Complementing FPT results, we show that PMS does not admit a polynomial kernel when parameterized by vertex cover number unless NP⊆coNP/poly. We also provide an exact exponential algorithm running in time O⁎(1.966n). Finally, considering graphs with structural assumptions, we show that PMS remains NP-hard on planar graphs.