  We introduce the factorization graph of a finite group and study its
connectedness and forbidden structures. We characterize all finite groups with
connected factorization graphs and classify those with connected bipartite
factorization graphs. Also, we obtain a classification of all groups with
claw-free, $K_{1,4}$-free, and square-free factorization graphs, excluding the
non-solvable groups in the latter case.
