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：对于图类$\mathcal{G}$，\textsc{子图补全到}$\mathcal{G}$问题的目标是：判断输入图$G$的顶点集是否存在子集$S$，使得通过对$S$诱导的子图进行补全操作修改$G$后，所得图属于$\mathcal{G}$。当$\mathcal{G}$为最小度数至少为$k$（$k$为常数）的图类时，我们得到了该问题的多项式时间算法，解决了Fomin等人（Algorithmica, 2020）提出的一个开放问题。当$\mathcal{G}$为不含任何$t+1$个顶点星图（对任意常数$t\geq 3$）和钻石图作为诱导子图的图类时，我们同样获得了该问题的多项式时间算法。这与Antony等人（Algorithmica, 2022）的结果形成对比：后者表明，对于每个常数$t\geq 5$，当$\mathcal{G}$为不含任何$t+1$个顶点星图作为诱导子图的图类时，该问题是NP完全的，且无法在次指数时间内求解（假设指数时间假说成立）。