# Collapsibility of non-cover complexes of graphs

## TL;DR

This paper proves that the non-cover complex of any graph is collapsible to a certain dimension, extending previous results from chordal graphs to all graphs, thus answering a question in topological combinatorics.

## Contribution

The paper generalizes the collapsibility property of non-cover complexes from chordal graphs to all graphs, confirming a conjecture by Aharoni.

## Key findings

- Non-cover complex of any graph is $(|V(G)|-i \, ext{γ}(G)-1)$-collapsible.
- Extends previous results from chordal graphs to all graphs.
- Provides a positive answer to Aharoni's question.

## Abstract

Given a graph $G$, the non-cover complex of $G$ is the combinatorial Alexander dual of the independence complex of $G$. Aharoni asked if the non-cover complex of a graph $G$ without isolated vertices is $(|V(G)|-i \gamma(G)-1)$-collapsible where $i \gamma(G)$ denotes the independent domination number of $G$. Extending a result by the second author, who verified Aharoni's question in the affirmative for chordal graphs, we prove that the answer to the question is yes for all graphs. Namely, we show that for a graph $G$, the non-cover complex of a graph $G$ is $(|V(G)|-i \gamma(G)-1)$-collapsible.