Open Access

Elisabeth Gaar, Dunja Pucher

• 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 Lovász theta function at the first level and including all exact subgraph constraints of subgraphs of order into the semidefinite program to compute the Lovász theta function at level . 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 Lovász 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 vertex-transitive 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 ofThe 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 Lovász theta function at the first level and including all exact subgraph constraints of subgraphs of order into the semidefinite program to compute the Lovász theta function at level . 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 Lovász 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 vertex-transitive 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 vertex-transitive ESH produces superior bounds compared to the ESH for Paley graphs.…