Shellability of complexes of directed trees

TL;DR

This paper investigates the shellability of complexes of directed trees, establishing conditions for shellability, characterizing specific cases like crosspolytopes, and analyzing the structure and decomposability of these complexes.

Contribution

It provides new insights into shellability criteria for complexes of directed trees, including cases with complete sources and double directed graphs, and explores their structural properties.

Findings

Complete source in a directed graph yields a straightforward shelling.

Only crosspolytopes among simplicial polytopes admit such shellings.

Complexes of directed trees in complete double directed graphs form unions of spheres.

Abstract

The question of shellability of complexes of directed trees was asked by R. Stanley. D. Kozlov showed that the existence of a complete source in a directed graph provides a shelling of its complex of directed trees. We will show that this property gives a shelling that is straightforward in some sense. Among the simplicial polytopes, only the crosspolytopes allow such a shelling. Furthermore, we show that the complex of directed trees of a complete double directed graph is a union of suitable spheres. We also investigate shellability of the maximal pure skeleton of a complex of directed trees. Also, we prove that is vertex-decomposable. For these complexes we describe the set of generating facets.