Extendability of $1$-decomposable complexes

TL;DR

This paper explores the extendability of 1-decomposable complexes, showing that such complexes can be extended to a larger skeleton while preserving 1-decomposability, thus generalizing previous results on shellable and 0-decomposable complexes.

Contribution

It proves that pure d-dimensional 1-decomposable complexes on n vertices can be extended to a larger skeleton, advancing understanding of complex extendability beyond shellability.

Findings

Pure d-dimensional 1-decomposable complexes can be extended to Δ_{n + d - 3}^{(d)}.

Extension preserves 1-decomposability during facet attachment.

Generalizes previous results on shellable and 0-decomposable complexes.

Abstract

A well-known conjecture of Simon (1994) states that any pure $d$-dimensional shellable complex on $n$ vertices can be extended to $\Delta_{n-1}^{(d)}$, the $d$-skeleton of the $(n-1)$-dimensional simplex, by attaching one facet at a time while maintaining shellability. The notion of $k$-decomposability for simplicial complexes, which generalizes shellability, was introduced by Provan and Billera (1980). Coleman, Dochtermann, Geist, and Oh (2022) showed that any pure $d$-dimensional $0$-decomposable complex on $n$ vertices can similarly be extended to $\Delta_{n-1}^{(d)}$, attaching one facet at a time while preserving $0$-decomposability. In this paper, we investigate the analogous question for $1$-decomposable complexes. We prove a slightly relaxed version: any pure $d$-dimensional $1$-decomposable complex on $n$ vertices can be extended to $\Delta_{n + d - 3}^{(d)}$, attaching one facet at a time while maintaining $1$-decomposability.