Foldings in graphs and relations with simplicial complexes and posets

TL;DR

This paper explores dismantlability in graphs and its connections to posets and simplicial complexes, establishing a correspondence between these frameworks and analyzing the implications for the polyhedral complex Hom(G,H).

Contribution

It provides a precise correspondence of dismantlability notions across graphs, posets, and simplicial complexes, and relates these to the structure of the Hom(G,H) complex.

Findings

Dismantlability in graphs corresponds to strong deformation retracts.

A triangle relating graphs, posets, and simplicial complexes clarifies their dismantlability relations.

Results refine understanding of the Hom(G,H) complex and its relation to graph foldings.

Abstract

We study dismantlability in graphs. In order to compare this notion to similar operations in posets (partially ordered sets) or in simplicial complexes, we prove that a graph G dismants on a subgraph H if and only if H is a strong deformation retract of G. Then, by looking at a triangle relating graphs, posets and simplicial complexes, we get a precise correspondence of the various notions of dismantlability in each framework. As an application, we study the link between the graph of morphisms from a graph G to a graph H and the polyhedral complex Hom(G,H); this gives a more precise statement about well known results concerning the polyhedral complex Hom(G,H) and its relation with foldings in G or H.