The Minimum Subgraph Complementation Problem

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.