  Let $G$ be a group and $Z(G)$ be its center. We associate a commuting graph
${\Gamma}(G)$, whose vertex set is $G\setminus Z(G)$ and two distinct vertices
are adjacent if they commute. We say that ${\Gamma}(G)$ is strong $k$ star free
if the $k$ star graph is not a subgraph of ${\Gamma}(G)$. In this paper, we
characterize all strong $5$ star free commuting graphs. As a byproduct, we
classify all strong claw-free graphs. Also, we prove that the set of all
non-abelian groups whose commuting graph is strong $k$ star free is finite.
