Exploring subgraph complementation to bounded degree graphs

TL;DR

This paper investigates the computational complexity of subgraph complementation problems, establishing NP-completeness for certain degree-bounded graph classes and fixed-parameter tractability when parameterized by degree.

Contribution

It answers an open question by proving NP-completeness for subgraph complementation to graphs with minimum degree at least k, and shows fixed-parameter tractability when parameterized by k.

Findings

NP-complete for minimum degree at least k when k is part of input

Fixed-parameter tractable for k-regular graphs parameterized by k

Addresses open problem in the complexity of subgraph complementation

Abstract

Graph modification problems are computational tasks where the goal is to change an input graph $G$ using operations from a fixed set, in order to make the resulting graph satisfy a target property, which usually entails membership to a desired graph class $\mathcal{C}$. Some well-known examples of operations include vertex-deletion, edge-deletion, edge-addition and edge-contraction. In this paper we address an operation known as subgraph complement. Given a graph $G$ and a subset $S$ of its vertices, the subgraph complement $G \oplus S$ is the graph resulting of complementing the edge set of the subgraph induced by $S$ in $G$. We say that a graph $H$ is a subgraph complement of $G$ if there is an $S$ such that $H$ is isomorphic to $G \oplus S$. For a graph class $\mathcal{C}$, subgraph complementation to $\mathcal{C}$ is the problem of deciding, for a given graph $G$, whether $G$ has a subgraph complement in $\mathcal{C}$. This problem has been studied and its complexity has been settled for many classes $\mathcal{C}$ such as $\mathcal{H}$-free graphs, for various families $\mathcal{H}$, and for classes of bounded degeneracy. In this work, we focus on classes graphs of minimum/maximum degree upper/lower bounded by some value $k$. In particular, we answer an open question of Antony et al. [Information Processing Letters 188, 106530 (2025)], by showing that subgraph complementation to $\mathcal{C}$ is NP-complete when $\mathcal{C}$ is the class of graphs of minimum degree at least $k$, if $k$ is part of the input. We also show that subgraph complementation to $k$-regular parameterized by $k$ is fixed-parameter tractable.