On subgraph complementation to H-free graphs

For a class \(\mathcal{G}\) of graphs, the problem SUBGRAPH COMPLEMENT TO \(\mathcal{G}\) asks whether one can find a subset \(S\) of vertices of the input graph \(G\) such that complementing the subgraph induced by \(S\) in \(G\) results in a graph in \(\mathcal{G}\). We investigate the complexity of the problem when \(\mathcal{G}\) is \(H\)-free for \(H\) being a complete graph, a star, a path, or a cycle. We obtain the following results: - When \(H\) is a \(K_t\) (a complete graph on \(t\) vertices) for any fixed \(t\geq 1\), the problem is solvable in polynomial-time. This applies even when \(\mathcal{G}\) is a subclass of \(K_t\)-free graphs recognizable in polynomial-time, for example, the class of \((t-2)\)-degenerate graphs. - When \(H\) is a \(K_{1,t}\) (a star graph on \(t+1\) vertices), we obtain that the problem is NP-complete for every \(t\geq 5\). This, along with known results, leaves only two unresolved cases - \(K_{1,3}\) and \(K_{1,4}\). - When \(H\) is a \(P_t\) (a path on \(t\) vertices), we obtain that the problem is NP-complete for every \(t\geq 7\), leaving behind only two unresolved cases - \(P_5\) and \(P_6\). - When \(H\) is a \(C_t\) (a cycle on \(t\) vertices), we obtain that the problem is NP-complete for every \(t\geq 8\), leaving behind four unresolved cases - \(C_4, C_5, C_6,\) and \(C_7\). Further, we prove that these hard problems do not admit subexponential-time algorithms (algorithms running in time \(2^{o(|V(G)|)}\)), assuming the Exponential Time Hypothesis. A simple complementation argument implies that results for \(\mathcal{G}\) are applicable for \(\overline{\mathcal{G}}\), thereby obtaining similar results for \(H\) being the complement of a complete graph, a star, a path, or a cycle. Our results generalize two main results and resolve one open question by Fomin et al. (Algorithmica, 2020).