On forbidden subgraphs of main supergraphs of groups

In this study, we explore the main supergraph $\mathcal{S}(G)$ of a finite group $G$, defined as an undirected, simple graph with a vertex set $G$ in which two distinct vertices, $a$ and $b$, are adjacent in $\mathcal{S}(G)$ if the order of one is a divisor of the order of the other. This is denoted as either $o(a)\mid o(b)$ or $o(b)\mid o(a)$, where $o(\cdot )$ is the order of an element. We classify finite groups for which the main supergraph is either a split graph or a threshold graph. Additionally, we characterize finite groups whose main supergraph is a cograph. Our classification extends to finite groups $G$ with $\mathcal{S}(G)$, a cograph that includes when $G$ is a direct product of two non-trivial groups, as well as when $G$ is either a dihedral group, a generalized quaternion group, a symmetric group, an alternating group, or a sporadic simple group.