Characterizing finite groups whose order supergraphs satisfy a connectivity condition

Let $\Gamma$ be an undirected and simple graph. A set $S$ of vertices in $\Gamma$ is called a {cyclic vertex cutset} of $\Gamma$ if $\Gamma - S$ is disconnected and has at least two components each containing a cycle. If $\Gamma$ has a cyclic vertex cutset, then it is said to be {cyclically separable}. For any finite group $G$, the order supergraph $\mathcal{S}(G)$ is the simple and undirected graph whose vertices are elements of $G$, and two vertices are adjacent if as elements of $G$ the order of one divides the order of the other. In this paper, we characterize the finite nilpotent groups and various non-nilpotent groups, such as the dihedral groups, the dicyclic groups, the EPPO groups, the symmetric groups, and the alternating groups, whose order supergraphs are cyclically separable.