A subgraph complement of the graph G is a graph obtained from G by complementing all the edges in one of its induced subgraphs. We study the following algorithmic question: for a given graph G and graph class $${\mathscr {G}}$$ G , is there a subgraph complement of G which is in $${\mathscr {G}}$$ G ? We show that this problem can be solved in polynomial time for various choices of the graphs class $${\mathscr {G}}$$ G , such as bipartite, d -degenerate, or cographs. We complement these results by proving that the problem is $${{\mathrm{NP}}}$$ NP -complete when $${\mathscr {G}}$$ G is the class of regular graphs.

中文翻译：

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子图补

图 G 的子图补是通过对 G 的其中一个诱导子图中的所有边进行补而从 G 获得的图。我们研究以下算法问题：对于给定的图 G 和图类 $${\mathscr {G}}$$ G ，在 $${\mathscr {G}}$$ G 中是否存在 G 的子图补集? 我们表明，对于图类 $${\mathscr {G}}$$ G 的各种选择，可以在多项式时间内解决这个问题，例如二部、d -退化或共图。我们通过证明当 $${\mathscr {G}}$$ G 是正则图类时问题是 $${{\mathrm{NP}}}$$ NP -complete 来补充这些结果。