Graph Square Roots of Small Distance from Degree One Graphs

Given a graph class \({\mathscr{H}}\), the task of the \({\mathscr{H}}\)-Square Root problem is to decide whether an input graph G has a square root H from \({\mathscr{H}}\). We are interested in the parameterized complexity of the problem for classes \({\mathscr{H}}\) 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 \(2^{2^{\mathcal {O}(k)}}\cdot n^{5}\). 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-kRoot 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-kRoot parameterized by k does not admit a subexponential kernel unless P = NP.