Characterizing finite groups whose order supergraphs satisfy a connectivity condition

TL;DR

This paper characterizes finite groups, including nilpotent, dihedral, dicyclic, EPPO, symmetric, and alternating groups, based on whether their order supergraphs are cyclically separable, revealing structural properties related to their divisibility relations.

Contribution

It provides a comprehensive characterization of finite groups whose order supergraphs are cyclically separable, including several important classes of groups.

Findings

Nilpotent groups with cyclically separable order supergraphs identified

Certain non-nilpotent groups like dihedral and symmetric groups characterized

Structural conditions for cyclic separability in order supergraphs established

Abstract

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.