For a class $\mathcal{G}$ of graphs, the problem SUBGRAPH COMPLEMENT TO $\mathcal{G}$ asks whether one can find a subset $S$ of vertices of the input graph $G$ such that complementing the subgraph induced by $S$ in $G$ results in a graph in $\mathcal{G}$. We investigate the complexity of the problem when $\mathcal{G}$ is $H$-free for $H$ being a complete graph, a star, a path, or a cycle. We obtain the following results: - When $H$ is a $K_t$ (a complete graph on $t$ vertices) for any fixed $t\geq 1$, the problem is solvable in polynomial-time. This applies even when $\mathcal{G}$ is a subclass of $K_t$-free graphs recognizable in polynomial-time, for example, the class of $(t-2)$-degenerate graphs. - When $H$ is a $K_{1,t}$ (a star graph on $t+1$ vertices), we obtain that the problem is NP-complete for every $t\geq 5$. This, along with known results, leaves only two unresolved cases - $K_{1,3}$ and $K_{1,4}$. - When $H$ is a $P_t$ (a path on $t$ vertices), we obtain that the problem is NP-complete for every $t\geq 7$, leaving behind only two unresolved cases - $P_5$ and $P_6$. - When $H$ is a $C_t$ (a cycle on $t$ vertices), we obtain that the problem is NP-complete for every $t\geq 8$, leaving behind four unresolved cases - $C_4, C_5, C_6,$ and $C_7$. Further, we prove that these hard problems do not admit subexponential-time algorithms (algorithms running in time $2^{o(|V(G)|)}$), assuming the Exponential Time Hypothesis. A simple complementation argument implies that results for $\mathcal{G}$ are applicable for $\overline{\mathcal{G}}$, thereby obtaining similar results for $H$ being the complement of a complete graph, a star, a path, or a cycle. Our results generalize two main results and resolve one open question by Fomin et al. (Algorithmica, 2020).

中文翻译：

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关于子图对无H图的补充

对于一类$ \ mathcal {G} $的图，问题SUBGRAPH COMPULEMENT TO $ \ mathcal {G} $的问题是，是否可以找到输入图$ G $的顶点的子集$ S $，从而补充所引出的子图$ S $在$ G $中的结果将在$ \ mathcal {G} $中产生图形。我们研究当$ \ mathcal {G} $无$ H $且$ H $是完整图形，星形，路径或循环时，问题的复杂性。我们得到以下结果：-对于任何固定的$ t \ geq 1 $，当$ H $是$ K_t $（在$ t $顶点上的完整图形）时，该问题可以在多项式时间内解决。即使$ \ mathcal {G} $是可在多项式时间内识别的无$ K_t $个图的子类，例如$（t-2）$退化图的类，这也适用。-当$ H $是$ K_ {1，t} $（$ t + 1 $个顶点上的星形图）时，我们得到的问题是每个$ t \ geq 5 $都是NP完全的。这，连同已知结果一起，只剩下两个未解决的案例-$ K_ {1,3} $和$ K_ {1,4} $。-当$ H $是$ P_t $（$ t $顶点上的路径）时，我们得到的问题是每个$ t \ geq 7 $都是NP完全的，只剩下两个未解决的情况-$ P_5 $和$ P_6 $。-当$ H $是$ C_t $（$ t $顶点上的循环）时，我们得到的问题是每个$ t \ geq 8 $都是NP完整的，剩下四个未解决的情况-$ C_4，C_5，C_6 ，$和$ C_7 $。进一步，我们证明了这些困难的问题不容许次指数时间算法（算法在时间$ 2 ^ {o（| V（G）|）} $中运行），假设是指数时间假说。一个简单的补充论证意味着$ \ mathcal {G} $的结果适用于$ \ overline {\ mathcal {G}} $，从而获得$ H $的相似结果是完整图，星形，路径或周期。我们的结果概括了两个主要结果，并解决了Fomin等人的一个未解决的问题。（Algorithmica，2020年）。