Parameterized Complexity of Perfectly Matched Sets

For an undirected graph G , a pair of vertex disjoint subsets p A, B q 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 p? k q ¨ n O p 1 q , and ii) K b,b -free graphs. We obtain a linear kernel for planar graphs and k O p d q -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. 2012 ACM Subject Classification Theory of computation Ñ Fixed parameter tractability