On subgraph complementation to H-free graphs

TL;DR

This paper investigates the computational complexity of the subgraph complementation problem for H-free graphs, providing polynomial-time algorithms for some cases and NP-completeness for others, with several unresolved cases remaining.

Contribution

It characterizes the complexity of subgraph complementation to H-free graphs for various H, resolving some open questions and establishing hardness results under ETH.

Findings

Polynomial-time solvability for H as a complete graph K_t.

NP-completeness for H as a star K_{1,t} for t≥5.

NP-completeness for H as a path P_t for t≥7.

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: - 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).