The independence graph of a finite group

Given a finite group $G,$ we denote by $Δ(G)$ 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 $Δ(G)$ is connected and classify the groups $G$ for which $Δ(G)$ is a planar graph.

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