Vertex Dismissibility and Scalability of Simplicial Complexes

TL;DR

This paper introduces new hierarchical classes of simplicial complexes extending vertex decomposability and shellability, linking topological properties with algebraic ideals and providing unified structural insights.

Contribution

It defines and explores strongly vertex dismissible, vertex dismissible, and scalable complexes, establishing their hierarchy and algebraic duals, with applications to classical combinatorial structures.

Findings

Strong vertex dismissibility implies vertex dismissibility.

Vertex dismissibility implies scalability.

Scalability implies initially Cohen-Macaulayness.

Abstract

We introduce and study strongly vertex dismissible, vertex dismissible, and scalable simplicial complexes as non-pure extensions of vertex decomposability and shellability. Strong vertex dismissibility is defined recursively by relaxing the shedding vertex condition, while vertex dismissibility and scalability are determined by the initial dimension skeleton. These classes form a strict hierarchy in which strong vertex dismissibility implies vertex dismissibility, which in turn implies scalability, and scalability implies initially Cohen-Macaulayness. On the algebraic side, we define strongly vertex divisible ideals, vertex divisible ideals, and ideals with degree quotients, and show that they are precisely the Alexander duals of the corresponding topological classes. This perspective yields a unified topological and homological structure together with skeletal characterizations that recover several classical results. For complexes of initial dimension one and the independence complexes of co-chordal and certain cycle graphs, this chain collapses to the purely combinatorial condition of weak connectedness.