Not all simplicial polytopes are weakly vertex-decomposable

TL;DR

This paper demonstrates that not all simplicial polytopes are weakly vertex-decomposable, providing the first examples of such polytopes, which challenges previous assumptions about their structure.

Contribution

It introduces the first known examples of simplicial polytopes that are not weakly 0-decomposable, expanding understanding of their combinatorial properties.

Findings

First examples of non-weakly 0-decomposable simplicial polytopes

Challenges previous assumptions about simplicial polytope decomposability

Provides insights into the structure and limitations of weak vertex-decomposability

Abstract

In 1980 Provan and Billera defined the notion of weak $k$-decomposability for pure simplicial complexes. They showed the diameter of a weakly $k$-decomposable simplicial complex $\Delta$ is bounded above by a polynomial function of the number of $k$-faces in $\Delta$ and its dimension. For weakly 0-decomposable complexes, this bound is linear in the number of vertices and the dimension. In this paper we exhibit the first examples of non-weakly 0-decomposable simplicial polytopes.