Closed neighborhood complexes of graphs

TL;DR

This paper explores the properties of the closed neighborhood complex of a graph, revealing its connections to other complexes and its fundamental group's relation to graph path homology.

Contribution

It establishes new links between the closed neighborhood complex and independence complexes, and characterizes its fundamental group in relation to path homology.

Findings

Connected the closed neighborhood complex with independence complexes.

Proved the fundamental group of the complex matches the graph's path homology group.

Provided new insights into the topological structure of graph complexes.

Abstract

The closed neighborhood complex $\mathcal{N}[G]$ of a simple graph $G$ is the simplicial complex whose simplices are finite sets of vertices contained in a closed neighborhood of a vertex in $G$. We reveal that the closed neighborhood complex has close connections with other concepts, including the independence complex of the canonical double covering and the independence complex of the neighborhood hypergraph. Furthermore, we show that the fundamental group of the closed neighborhood complex is isomorphic to Grigor'yan--Lin--Muranov--Yau's fundamental group of a graph introduced in the study of path homology.