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.

翻译：图 $G$ 在顶点 $v$ 上的局部补是一种运算，其结果是一个新图 $G*v$，其中 $v$ 的邻域被取补。该运算在图论和量子计算领域已被广泛研究。本文引入了局部补问题，这是一个刻画应用一系列局部补运算复杂性的判定问题。给定一个图 $G$、一个顶点序列 $s$ 以及一对顶点 $u,v$，该问题询问在根据 $s$ 应用局部补运算后得到的图中，边 $(u,v)$ 是否存在。本工作的主要贡献是证明了该问题是 $\mathsf{P}$ 完全的，这意味着计算一系列局部补运算不太可能被高效地并行化。证明基于从著名的 $\mathsf{P}$ 完全问题——电路值问题的归约，通过局部补运算来模拟电路。此外，本文还分析了该问题在不同限制条件下的复杂性。特别地，证明了对于完全图和星形图，该问题属于 $\mathsf{LOGSPACE}$。最后，本文推测该问题对于圆图类仍然是 $\mathsf{P}$ 完全的。