Vertex orders in higher dimensions

TL;DR

This paper explores the properties of higher-dimensional interval complexes, demonstrating conditions under which they are shellable and vertex decomposable, and providing examples of non-shellable complexes in higher dimensions.

Contribution

It establishes that strongly connected unit interval complexes are shellable and vertex decomposable, and presents examples of non-shellable interval complexes in higher dimensions.

Findings

Strongly connected unit interval complexes are shellable.

Such complexes are also vertex decomposable.

Examples of non-shellable interval complexes in dimensions two and higher.

Abstract

Unit interval and interval complexes are higher-dimensional generalizations of unit interval and interval graphs, respectively. We show that strongly connected unit interval complexes are shellable with shellings induced by their unit interval orders. We also show that these complexes are vertex decomposable and hence shelling completable. On the other hand, we give simple examples of strongly connected interval complexes that are not shellable in dimensions two and higher.