Abstract
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:
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When H is a \(K_t\) (a complete graph on t vertices) for any fixed \(t\ge 1\), 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.
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When H is a \(K_{1,t}\) (a star graph on \(t+1\) vertices), we obtain that the problem is NP-complete for every \(t\ge 5\). This, along with known results, leaves only two unresolved cases - \(K_{1,3}\) and \(K_{1,4}\).
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When H is a \(P_t\) (a path on t vertices), we obtain that the problem is NP-complete for every \(t\ge 7\), leaving behind only two unresolved cases - \(P_5\) and \(P_6\).
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When H is a \(C_t\) (a cycle on t vertices), we obtain that the problem is NP-complete for every \(t\ge 8\), 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).
Partially supported by SERB Grant SRG/2019/002276: “Complexity Dichotomies for Graph Modification Problems”.
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Antony, D., Garchar, J., Pal, S., Sandeep, R.B., Sen, S., Subashini, R. (2021). On Subgraph Complementation to H-free Graphs. In: Kowalik, Ł., Pilipczuk, M., Rzążewski, P. (eds) Graph-Theoretic Concepts in Computer Science. WG 2021. Lecture Notes in Computer Science(), vol 12911. Springer, Cham. https://doi.org/10.1007/978-3-030-86838-3_9
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