Vertex-minimal graphs and fixing sets for modular p-groups

For a finite group [Formula: see text], define [Formula: see text] as the minimum number of vertices among all graphs [Formula: see text] such that [Formula: see text]. In this paper, we compute the order of vertex-minimal graphs with an automorphism group isomorphic to a modular [Formula: see text]-group. The fixing number of a graph [Formula: see text] is defined as the smallest number of vertices in [Formula: see text] that, when fixed, makes [Formula: see text] trivial. For a finite group [Formula: see text], the fixing set is defined as the set of all fixing numbers of graphs having automorphism groups isomorphic to [Formula: see text]. We show that any graph [Formula: see text] whose automorphism group is a modular [Formula: see text]-group has the fixing number 1. As a result, the modular [Formula: see text]-group’s fixing set becomes [Formula: see text].