In the literature it is known that the decomposable numerical range W ∧ k (A) of A ∈ C n×n is not necessarily convex. But it is not known whether W ∧ k (A) is star-shaped. We construct a symmetric unitary matrix A ∈ C n×n such that the decomposable numerical range W ∧ k (A) is not star-shaped and hence not simply connected. We then consider a real analog R ∧ k (A) and show that R ∧ k (A) is star-shaped if A ∈ C n×n is skew symmetric. Such star-shapedness result is also true for the Pfaffian numerical range P ∧ k (A).