Homology, Homotopy and Applications

Volume 25 (2023)

Number 1

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

Pages: 331 – 343

DOI: https://dx.doi.org/10.4310/HHA.2023.v25.n1.a17


Yu Tajima (Department of Mathematics, Graduate School of Science, Hokkaido University, Kita-ku, Sapporo, Japan)

Masahiko Yoshinaga (Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Japan)


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 $2$.


magnitude homology, graph, discrete Morse theory

2010 Mathematics Subject Classification

05C10, 05E45, 55N35

Received 12 October 2021

Received revised 26 April 2022

Accepted 1 June 2022

Published 26 April 2023