The independence graph of a finite group

Abstract

Given a finite group $G,$ we denote by $\Delta(G)$ the graph whose vertices are the elements $G$ and where two vertices $x$ and $y$ are adjacent if there exists a minimal generating set of $G$ containing $x$ and $y.$ We prove that $\Delta(G)$ is connected and classify the groups $G$ for which $\Delta(G)$ is a planar graph.