Minimal graphs for contractible and dismantlable properties

TL;DR

This paper investigates subclasses of contractible graphs that can be simplified using specific moves, constructs minimal examples distinguishing these classes, and relates them to dismantlable and collapsible complexes, addressing previous misconceptions.

Contribution

It introduces and analyzes subclasses of contractible graphs based on restricted moves, providing minimal examples and correcting prior inaccuracies in the literature.

Findings

Constructed minimal examples of graph classes with different collapsibility properties.

Established relationships between contractible, dismantlable, and collapsible graphs.

Identified a minimal counterexample to a previous erroneous claim.

Abstract

The notion of a contractible transformation on a graph was introduced by Ivashchenko as a means to study molecular spaces arising from digital topology and computer image analysis, and more recently has been applied to topological data analysis. Contractible transformations involve a list of four elementary moves that can be performed on the vertices and edges of a graph, and it has been shown by Chen, Yau, and Yeh that these moves preserve the simple homotopy type of the underlying clique complex. A graph is said to be ${\mathcal I}$-contractible if one can reduce it to a single isolated vertex via a sequence of contractible transformations. Inspired by the notions of collapsible and non-evasive simplicial complexes, in this paper we study certain subclasses of ${\mathcal I}$-contractible graphs where one can collapse to a vertex using only a subset of these moves. Our main results involve constructions of minimal examples of graphs for which the resulting classes differ. We also relate these classes of graphs to the notion of $k$-dismantlable graphs and $k$-collapsible complexes, which also leads to a minimal counterexample to an erroneous claim of Ivashchenko from the literature. We end with some open questions.