Perfectly Matched Sets in Graphs: Hardness, Kernelization Lower Bound, and FPT and Exact Algorithms
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In an undirected graph G=(V,E), we say (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 NP-hard on planar graphs and W[1]-hard when parameterized by solution size k even when restricted to split graphs and bipartite graphs. We show that PMS parameterized by vertex cover number does not admit a polynomial kernel unless NP⊆ coNP/poly. We give FPT algorithms with respect to the parameters distance to cluster, distance to co-cluster and treewidth. We also provide an exact exponential algorithm running in time O^*(1.964^n).
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