Graph Square Roots of Small Distance from Degree One Graphs

Abstract

Given a graph class \(\mathcal {H}\), the task of the \(\mathcal {H}\)-Square Root problem is to decide, whether an input graph G has a square root H that belongs to \(\mathcal {H}\). We are interested in the parameterized complexity of the problem for classes \(\mathcal {H}\) 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 providing algorithms with running time \(2^{2^{\mathcal {O}(k)}}\cdot n^{\mathcal {O}(1)}\). 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 \(2^{2^{o(k)}}\cdot n^{\mathcal {O}(1)}\) unless the Exponential Time Hypothesis fails. Moreover, we point out that VC- \(k\) Root parameterized by k does not admit a subexponential kernel unless P=N P.