$\mathsf{P}$-completeness of Graph Local Complementation

Abstract: Local complementation of a graph $G$ on vertex $v$ is an operation that results in a new graph $G*v$, where the neighborhood of $v$ is complemented. This operation has been widely studied in graph theory and quantum computing. This article introduces the Local Complementation Problem, a decision problem that captures the complexity of applying a sequence of local complementations. Given a graph $G$, a sequence of vertices $s$, and a pair of vertices $u,v$, the problem asks whether the edge $(u,v)$ is present in the graph obtained after applying local complementations according to $s$. The main contribution of this work is proving that this problem is $\mathsf{P}$-complete, implying that computing a sequence of local complementation is unlikely to be efficiently parallelizable. The proof is based on a reduction from the Circuit Value Problem, a well-known $\mathsf{P}$-complete problem, by simulating circuits through local complementations. Aditionally, the complexity of this problem is analyzed under different restrictions. In particular, it is shown that for complete and star graphs, the problem belongs to $\mathsf{LOGSPACE}$. Finally, it is conjectured that the problem remains $\mathsf{P}$-complete for the class of circle graphs.