Exploring subgraph complementation to bounded degree graphs

TL;DR

The paper studies subgraph complementation as a graph modification operation and analyzes its complexity when targeting graphs with bounded degree. It proves NP-completeness for transforming a graph to have maximum degree at most $k$ (with $k$ part of the input) and establishes FPT results for transforming to $k$-regular graphs and, via parameterization by $k$, for maximum-degree targets. It also provides a 3-approximation for minimizing the resulting maximum degree and discusses related approximations and kernelization aspects. These results advance understanding of degree-constrained modifications and open paths to broader parameterized and approximation approaches in subgraph complementation problems.

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.

Figures (1)

• Figure 1: Subfigure $(a)$ illustrates the graph reduction used in the proof of Theorem \ref{['thm:min_deg_NPc']} (the graph $G'$). Except for the circle corresponding to $G$, all circles are complete subgraphs. There are all possible edges between $G$ and $K_t$, and between $K_t$ and $K_s$. Vertices in $K_a^v$ (resp. $K_b^u$) are adjacent to a single vertex $v$ in $K_t$ (resp. $u$ in $K_s$. We state below the name of every subgraph the (maximum) degree $\Delta$ of its vertices. Subfigure $(b)$ shows the vertices of the set $S$ highlighted in gray, as well as the maximum degree of the subgraphs in $G' \oplus S$.

Theorems & Definitions (18)

• Theorem 3.1
• proof
• Remark 3.1
• Proposition 4.1
• proof
• Lemma 4.1
• proof
• Theorem 4.1
• proof
• Proposition 4.2