The path-missing and path-free complexes of a directed graph

TL;DR

This paper investigates two simplicial complexes derived from a directed graph related to vertices s and t, demonstrating they have simple homotopy types using discrete Morse theory.

Contribution

It introduces and analyzes the path-missing and path-free complexes, revealing their homotopy types are either contractible or sphere-like, using discrete Morse theory.

Findings

Both complexes are either contractible or homotopy equivalent to spheres.

The complexes have well-behaved homotopy types.

Discrete Morse theory is used to analyze their topological properties.

Abstract

We study two simplicial complexes arising from a directed graph $G = (V, E)$ with two chosen vertices $s$ and $t$: the *path-free complex*, consisting of all subsets $F \subseteq E$ that contain no path from $s$ to $t$, and the *path-missing complex*, its Alexander dual. Using discrete Morse theory, we prove that both complexes have well-behaved homotopy types -- either contractible or homotopy-equivalent to spheres.