Partial complementation of graphs

TL;DR

This paper investigates the algorithmic problem of determining whether a partial complement of a given graph belongs to specific graph classes, providing polynomial-time solutions for some classes and NP-completeness for others.

Contribution

It introduces the problem of partial graph complementations and characterizes its computational complexity across various graph classes, offering new polynomial-time algorithms and NP-completeness results.

Findings

Polynomial-time algorithms for bipartite, degenerate, and cographs.

NP-completeness for the class of r-regular graphs.

Extension of graph complement concepts to partial complementations.

Abstract

A partial complement of the graph $G$ is a graph obtained from $G$ by complementing all the edges in one of its induced subgraphs. We study the following algorithmic question: for a given graph $G$ and graph class $\mathcal{G}$, is there a partial complement of $G$ which is in $\mathcal{G}$? We show that this problem can be solved in polynomial time for various choices of the graphs class $\mathcal{G}$, such as bipartite, degenerate, or cographs. We complement these results by proving that the problem is NP-complete when $\mathcal{G}$ is the class of $r$-regular graphs.