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On subgraph complementation to Hfree graphs
For a class π’ of graphs, the problem SUBGRAPH COMPLEMENT TO π’ 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 π’. We investigate the complexity of the problem when π’ is Hfree 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 tβ₯ 1, the problem is solvable in polynomialtime. This applies even when π’ is a subclass of K_tfree graphs recognizable in polynomialtime, for example, the class of (t2)degenerate graphs.  When H is a K_1,t (a star graph on t+1 vertices), we obtain that the problem is NPcomplete for every tβ₯ 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 NPcomplete for every tβ₯ 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 NPcomplete for every tβ₯ 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 subexponentialtime algorithms (algorithms running in time 2^o(V(G))), assuming the Exponential Time Hypothesis. A simple complementation argument implies that results for π’ are applicable for π’, 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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