The Minimum Subgraph Complementation Problem
This paper studies the Minimum Subgraph Completion problem: given a graph $G$ and a target graph class $mathcal{C}$, find a smallest vertex subset $S$ such that the completion of the induced subgraph $G[S]$ belongs to $mathcal{C}$. We establish, for the first time, a systematic framework for polynomial-time solvability of this problem. Our approach resolves several nontrivial transformations—including bipartite/co-bipartite/split graph interconversions, regular bipartite graphs to chordal graphs, forests to fixed degenerate graph classes, and disconnected/2-connected graph conversions. Methodologically, we integrate structural graph analysis, modular decomposition, matching theory, and degeneracy-order-based dynamic programming to design compact, scalable, problem-specific algorithms. Our results fill a fundamental theoretical gap in polynomial-time tractability for subgraph completion optimization and provide the first unified algorithmic paradigm for graph class transformation.