Graph Square Roots of Small Distance from Degree One Graphs
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Given a graph class ℋ, the task of the ℋ-Square Root problem is to decide, whether an input graph G has a square root H from ℋ. We are interested in the parameterized complexity of the problem for classes ℋ that are composed by the graphs at vertex deletion distance at most k from graphs of maximum degree at most one, that is, we are looking for a square root H such that there is a modulator S of size k such that H-S is the disjoint union of isolated vertices and disjoint edges. We show that different variants of the problems with constraints on the number of isolated vertices and edges in H-S are FPT when parameterized by k by demonstrating algorithms with running time 2^2^O(k)· n^O(1). We further show that the running time of our algorithms is asymptotically optimal and it is unlikely that the double-exponential dependence on k could be avoided. In particular, we prove that the VC-k Root problem, that asks whether an input graph has a square root with vertex cover of size at most k, cannot be solved in time 2^2^o(k)· n^O(1) unless Exponential Time Hypothesis fails. Moreover, we point out that VC-k Root parameterized by k does not admit a subexponential kernel unless P=NP.
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