Subgraph complementation is an operation that toggles all adjacencies inside a selected vertex set. Given a graph \(G\) and a target class \(\mathcal{C}\), the Minimum Subgraph Complementation problem asks for a minimum-size vertex set \(S\) such that complementing the subgraph induced by \(S\) transforms \(G\) into a graph belonging to \(\mathcal{C}\). While the decision version of Subgraph Complementation has been extensively studied and is NP-complete for many graph classes, the algorithmic complexity of its optimization variant has remained largely unexplored. In this paper, we study MSC from an algorithmic perspective. We present polynomial-time algorithms for MSC in several nontrivial settings. Our results include polynomial-time solvability for transforming graphs between bipartite, co-bipartite, and split graphs, as well as for complementing bipartite regular graphs into chordal graphs. We also show that MSC to the class of graphs of fixed degeneracy can be solved in polynomial time when the input graph is a forest. Moreover, we investigate MSC with respect to connectivity and prove that MSC to the class of disconnected graphs and to the class of 2-connected graphs can be solved in polynomial time for arbitrary inputs.

翻译：子图补全是一种操作，它切换选定顶点集内部的所有邻接关系。给定图 \(G\) 和目标图类 \(\mathcal{C}\)，最小子图补全问题要求找到一个最小规模的顶点集 \(S\)，使得对由 \(S\) 诱导的子图进行补全操作后，将 \(G\) 转换为属于 \(\mathcal{C}\) 的图。虽然子图补全的判定版本已被广泛研究，并且对于许多图类而言是 NP 完全的，但其优化变体的算法复杂性在很大程度上仍未得到探索。在本文中，我们从算法角度研究最小子图补全问题。我们针对若干非平凡场景提出了多项式时间算法。我们的结果包括：在二分图、共二分图和分裂图之间转换图的多项式时间可解性，以及将二分正则图补全为弦图的多项式时间可解性。我们还证明了，当输入图为森林时，将图转换为固定退化度图类的最小子图补全问题可在多项式时间内求解。此外，我们研究了与连通性相关的最小子图补全问题，并证明了对于任意输入，将图转换为不连通图类以及 2-连通图类的最小子图补全问题均可在多项式时间内求解。