Algorithms for subgraph complementation to some classes of graphs

Abstract

For a class $\mathcal{G}$ of graphs, the objective of \textsc{Subgraph Complementation to} $\mathcal{G}$ is to find whether there exists a subset $S$ of vertices of the input graph $G$ such that modifying $G$ by complementing the subgraph induced by $S$ results in a graph in $\mathcal{G}$. We obtain a polynomial-time algorithm for the problem when $\mathcal{G}$ is the class of graphs with minimum degree at least $k$, for a constant $k$, answering an open problem by Fomin et al. (Algorithmica, 2020). When $\mathcal{G}$ is the class of graphs without any induced copies of the star graph on $t+1$ vertices (for any constant $t\geq 3$) and diamond, we obtain a polynomial-time algorithm for the problem. This is in contrast with a result by Antony et al. (Algorithmica, 2022) that the problem is NP-complete and cannot be solved in subexponential-time (assuming the Exponential Time Hypothesis) when $\mathcal{G}$ is the class of graphs without any induced copies of the star graph on $t+1$ vertices, for every constant $t\geq 5$.