vertex decomposability of complexes associated to forests

In this article, we discuss the vertex decomposability of three well-studied simplicial complexes associated to forests. in particular, we show that the bounded degree complex of a forest and the complex of directed trees of a multidiforest is vertex decomposable. we then prove that the non-cover complex of a forest is either contractible or homotopy equivalent to a sphere. finally we provide a complete characterization of forests whose non-cover complexes are vertex decomposable.