Combinatorial and Topological Aspects of Path Posets, and Multipath Cohomology

TL;DR

This paper explores the combinatorial and topological properties of path posets in directed graphs, introduces computational tools for multipath cohomology, and investigates their realizability as face posets of simplicial complexes.

Contribution

It develops acyclicity criteria, computes multipath cohomology for linear graphs, and interprets path posets as face posets, advancing understanding of multipath cohomology.

Findings

Acyclicity criteria for path posets

Computed multipath cohomology groups for oriented linear graphs

Interpreted path posets as face posets of simplicial complexes

Abstract

Multipath cohomology is a cohomology theory for directed graphs, which is defined using the path poset. The aim of this paper is to investigate combinatorial properties of path posets, and to provide computational tools for multipath cohomology. In particular, we develop acyclicity criteria, and provide computations of multipath cohomology groups of oriented linear graphs. We further interpret the path poset as the face poset of a simplicial complex, and we investigate realisability problems.