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

翻译：对于一类 $\mathcal{G}$ 的图，\textsc{Subgraph Complementation to} $\mathcal{G}$ 的目的是找到输入图 $G$ 的一个顶点子集 $S$，使得通过对由 $S$ 引出的亚图求补得到的图属于 $\mathcal{G}$ 类。我们获得了一个多项式时间算法，当 $\mathcal{G}$ 是最小度数至少为 $k$ 的图类时，其中 $k$ 是一个常数。这个结果回答了Fomin等人提出的一个开放问题 （Fomin et al.，Algorithmica，2020）。当 $\mathcal{G}$ 不包含星形图（任意常数 $t\geq 3$ 的顶点数）和菱形图的诱导子图时，我们获得了一个多项式时间算法。这与Antony等人的结果形成了对比（Antony et al.，Algorithmica，2022），他们证明了当 $\mathcal{G}$ 是没有诱导 $t+1$ 个顶点的星形图（$t\geq 5$）时，该问题是 NP-hard 的，并且不能以次指数时间解决（假设指数时间假设成立）。