Algorithms for subgraph complementation to some classes of graphs

Abstract

For a class $G$ of graphs, the objective of Subgraph Complementation to $G$ 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 $G$. We obtain a polynomial-time algorithm for the problem when $G$ 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 $G$ is the class of graphs without any induced copies of the star graph on $t+1$ vertices (for any constant $t\ge 3$) 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 $G$ is the class of graphs without any induced copies of the star graph on $t+1$ vertices, for every constant $t\ge 5$.