On Subgraph Complementation to H-free Graphs

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 $t\ge 1$, 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 d-degenerate graphs, where $d=t-2$ for $t>2$.

• 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 the 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 $t\ge 7$, 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 $t\ge 7$, leaving behind only three unresolved cases—$C_4, C_5,$ and $C_6$.

Further, we prove that these hard problems do not admit subexponential-time algorithms (algorithms running in time $2^{o(\mid V(G)\mid )}$), assuming the Exponential Time Hypothesis. We show that the complexity results on a graph class $\mathcal {G}$ is also true for the class $\overline{\mathcal {G}}$ of the complement graphs of $\mathcal {G}$. Therefore, each of the above results mentioned for the H-free class of graphs is also valid for the $\overline{H}$-free class of graphs. It is noteworthy that our results generalize two main results, namely, Subgraph Complement to triangle-free graphs and Subgraph Complement to d-degenerate graphs, and resolves one open question due to Fomin et al. (Algorithmica, 2020).