  The purpose of this note is to define a graph whose vertex set is a finite
group $G$, whose edge set is contained in that of the commuting graph of $G$
and contains the enhanced power graph of $G$. We call this graph the deep
commuting graph of $G$. Two elements of $G$ are joined in the deep commuting
graph if and only if their inverse images in every central extension of $G$
commute.
  We give conditions for the graph to be equal to either of the enhanced power
graph and the commuting graph, and show that the automorphism group of $G$ acts
as automorphisms of the deep commuting graph.
