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

Darij Grinberg; Lukas Katthän; Joel Brewster Lewis

doi:10.48550/arxiv.2102.07894

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

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

Darij Grinberg, Lukas Katthän and Joel Brewster Lewis

arXiv.org

15 Feb 2021

DOI: https://doi.org/10.48550/arxiv.2102.07894

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Abstract

Mathematics - Algebraic Topology

Mathematics - Combinatorics

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.

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Title: The path-missing and path-free complexes of a directed graph
Creators: Darij Grinberg
Lukas Katthän
Joel Brewster Lewis
Publication Details: arXiv.org
Resource Type: Preprint
Language: English
Academic Unit: Mathematics
Other Identifier: 991021862235204721

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

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Abstract

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