The stability number of a graph, defined as the cardinality of the largest set of pairwise non-adjacent vertices, is NP-hard to compute. The exact subgraph hierarchy (ESH) provides a sequence of increasingly tighter upper bounds on the stability number, starting with the Lov\'asz theta function at the first level and including all exact subgraph constraints of subgraphs of order $k$ into the semidefinite program to compute the Lov\'asz theta function at level $k$. In this paper, we investigate the ESH for Paley graphs, a class of strongly regular, vertex-transitive graphs. We show that for Paley graphs, the bounds obtained from the ESH remain the Lov\'asz theta function up to a certain threshold level, i.e., the bounds of the ESH do not improve up to a certain level. To overcome this limitation, we introduce the local ESH for the stable set problem for vertex-transitive graphs such as Paley graphs. We prove that this new hierarchy provides upper bounds on the stability number of vertex-transitive graphs that are at least as tight as those obtained from the ESH. Additionally, our computational experiments reveal that the local ESH produces superior bounds compared to the ESH for Paley graphs.

翻译：图的稳定数定义为最大两两不相邻顶点集的基数，其计算是NP难的。精确子图层次结构（ESH）提供了一系列关于稳定数的渐紧上界：第一级从Lovász theta函数开始，第k级通过将所有k阶子图的精确子图约束纳入计算Lovász theta函数的半定规划中。本文研究了Paley图（一类强正则的顶点传递图）的ESH。我们证明对于Paley图，ESH所获得的界在达到特定阈值级前始终为Lovász theta函数，即ESH的界在特定层级前不会改进。为克服此局限，我们针对Paley图等顶点传递图的稳定集问题提出了局部ESH。我们证明该新层次结构为顶点传递图稳定数提供的上界至少与ESH所得界同样紧致。此外，计算实验表明对于Paley图，局部ESH产生的界优于ESH。