# On subgraph complementation to H-free graphs

For a class 𝒢 of graphs, the problem SUBGRAPH COMPLEMENT TO 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 𝒢. We investigate the complexity of the problem when 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≥ 1, the problem is solvable in polynomial-time. This applies even when 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≥ 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≥ 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≥ 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 𝒢 are applicable for 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).

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