  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 $2$.
