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.

翻译：图修改问题是一类计算任务，其目标是通过固定操作集中的操作改变输入图$G$，使得结果图满足目标性质，通常意味着其属于某个期望的图类$\mathcal{C}$。一些著名的操作包括顶点删除、边删除、边添加和边收缩。本文研究一种称为子图补的操作。给定图$G$及其顶点子集$S$，子图补$G \oplus S$是通过对$G$中由$S$诱导的子图的边集取补得到的图。若存在$S$使得图$H$与$G \oplus S$同构，则称$H$是$G$的子图补。对于图类$\mathcal{C}$，子图补到$\mathcal{C}$的问题是指：对于给定图$G$，判断$G$是否存在属于$\mathcal{C}$的子图补。该问题已被广泛研究，其计算复杂度已在许多图类$\mathcal{C}$中得到确定，例如针对不同族$\mathcal{H}$的$\mathcal{H}$-free图，以及有界退化图类。本文重点关注最小/最大度上/下界为某值$k$的图类。特别地，我们通过证明当$\mathcal{C}$为最小度至少为$k$的图类且$k$作为输入的一部分时，子图补到$\mathcal{C}$的问题是NP完全的，从而回答了Antony等人[Information Processing Letters 188, 106530 (2025)]的开放性问题。我们还证明了以$k$为参数的子图补到$k$-正则图问题是固定参数可解的。