On the minimum cut-sets of the power graph of a finite cyclic group

TL;DR

This paper investigates the minimum cut-sets of the power graph of finite cyclic groups, extending previous characterizations to cases with four or more prime divisors by identifying specific cut-sets.

Contribution

It extends the characterization of minimum cut-sets in power graphs of cyclic groups to cases with at least four prime divisors, identifying key cut-sets.

Findings

Characterization of certain cut-sets for r ≥ 4

Identification of all minimum cut-sets in these cases

Extension of previous results for r ≤ 3

Abstract

The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple graph with vertex set $G$, in which two distinct vertices are adjacent if one of them is a power of the other. For an integer $n\geq 2$, let $C_n$ denote the cyclic group of order $n$ and let $r$ be the number of distinct prime divisors of $n$. The minimum cut-sets of $\mathcal{P}(C_n)$ are characterized in \cite{cps} for $r\leq 3$. In this paper, for $r\geq 4$, we identify certain cut-sets of $\mathcal{P}(C_n)$ such that any minimum cut-set of $\mathcal{P}(C_n)$ must be one of them.