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Multipath complexes of bidirectional polygonal digraphs

In this work we study the homotopy type of multipath complexes of bidirectional path graphs and polygons, motivated by works of Vrećica and Živaljević 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.