Local sets, a graph structure invariant under local complementation, have been originally introduced in the context of quantum computing for the study of quantum entanglement within the so-called graph state formalism. A local set in a graph is made of a non-empty set of vertices together with its odd neighborhood. We show that any graph can be covered by minimal local sets, i.e. that every vertex is contained in at least one local set that is minimal by inclusion. More precisely, we introduce an algorithm for finding a minimal local set cover in polynomial time. This result is proved by exploring the link between local sets and cut-rank. We prove some additional results on minimal local sets: we give tight bounds on their size, and we show that there can be exponentially many of them in a graph. Finally, we provide an extension of our definitions and our main result to $q$-multigraphs, the graphical counterpart of quantum qudit graph states.

翻译：局部集是在局部补变换下不变的图结构，最初在量子计算领域被引入，用于研究所谓图态形式化中的量子纠缠。图中的局部集由非空顶点集及其奇邻域组成。我们证明，任何图都可以被最小局部集覆盖，即每个顶点至少包含在一个按包含关系为最小的局部集中。更准确地，我们提出了一种多项式时间内寻找最小局部集覆盖的算法。这一结果通过探索局部集与割秩之间的联系得证。我们进一步得到了关于最小局部集的性质：给出了其大小的紧界，并证明图中可能存在指数数量的最小局部集。最后，我们将定义和主要结论推广至$q$-重图——即量子qudit图态的图论对应物。