Vertex decomposability of complexes associated to forests

TL;DR

This paper investigates the vertex decomposability of various simplicial complexes related to forests, establishing new results on their topological properties and providing characterizations for specific cases.

Contribution

It proves vertex decomposability for the bounded degree complex and directed trees complex of forests, and characterizes forests with vertex decomposable non-cover complexes.

Findings

Bounded degree complex of a forest is vertex decomposable.

Complex of directed trees of a multidiforest is vertex decomposable.

Non-cover complex of a forest is either contractible or homotopy equivalent to a sphere.

Abstract

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 are 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.