The invariably generating graph of the alternating and symmetric groups

Given a finite group $G$, the invariably generating graph of $G$ is defined as the undirected graph in which the vertices are the nontrivial conjugacy classes of $G$, and two classes are adjacent if and only if they invariably generate $G$. In this paper we study this object for alternating and symmetric groups. The main result of the paper states that, if we remove the isolated vertices from the graph, the resulting graph is connected and has diameter at most $6$.