Magnitude homology of graphs and discrete Morse theory on Asao-Izumihara complexes

TL;DR

This paper explores the homotopy type of Asao-Izumihara complexes related to graph magnitude homology, proving they are homotopy equivalent to wedges of spheres for certain graphs and identifying new diagonal graphs.

Contribution

It establishes the homotopy equivalence of Asao-Izumihara complexes to wedges of spheres for pawful graphs and introduces a generalized concept of pawful graphs.

Findings

Asao-Izumihara complexes are homotopy equivalent to wedges of spheres for pawful graphs.

Identified new non-pawful diagonal graphs of diameter 2.

Connected homotopy type results with the diagonality of magnitude homology groups.

Abstract

Recently, Asao and Izumihara introduced CW-complexes whose homology groups are isomorphic to direct summands of the graph magnitude homology group. In this paper, we study the homotopy type of the CW-complexes in connection with the diagonality of magnitude homology groups. We prove that the Asao-Izumihara complex is homotopy equivalent to a wedge of spheres for pawful graphs introduced by Y. Gu. The result can be considered as a homotopy type version of Gu's result. We also formulate a slight generalization of the notion of pawful graphs and find new non-pawful diagonal graphs of diameter $2$.