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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. Two graph are locally equivalent if on can be reached from the other one through local complementation. It was previously established that recognizing locally equivalent graphs can be done in $\mathcal{O}(n^4)$ time. We sharpen this result by proving it can be decided in $\mathcal{O}(\log^2(n))$ parallel time with $n^{\mathcal{O}(1)}$ processors. As a second contribution, we introduce 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 $s$. Regardless it simplicity, it is proven to be $\mathsf{P}$-complete, therefore it is unlikely to be efficiently parallelizable. Finally, it is conjectured that Local Complementation Problem remains $\mathsf{P}$-complete when restricted to circle graphs.