Multipath complexes of bidirectional polygonal digraphs

TL;DR

This paper investigates the topological properties of multipath complexes in bidirectional polygonal digraphs, revealing their homotopy types, connectivity, and homology, with implications for graph theory and algebraic topology.

Contribution

It characterizes the homotopy types of multipath complexes for bidirectional path and polygonal digraphs, extending previous work on cycle-free complexes and introducing new topological insights.

Findings

Bidirectional path graphs are homotopic to spheres.

Multipath complexes of bidirectional polygons are highly connected.

Homology groups are computed using Mayer-Vietoris spectral sequence.

Abstract

In this work we study the homotopy type of multipath complexes of bidirectional path graphs and polygons, motivated by works of Vre\'cica and \v{Z}ivaljevi\'c on cycle-free chessboard complexes (that is, multipath complexes of complete digraphs). In particular, we show that bidirectional path graphs are homotopic to spheres and that, in analogy with cycle-free chessboard complexes, multipath complexes of bidirectional polygonal digraphs are highly connected. Using a Mayer-Vietoris spectral sequence, we provide a computation of the associated homology groups. We study T-operations on graphs, and show that this corresponds to taking suspensions of multipath complexes. We further discuss (non) shellability properties of such complexes, and present new open questions.