Cofibration category structures on the category of graphs

In this article, we show that there is no cofibration category structure on the category of finite graphs with $\times$-homotopy equivalences as the class of weak equivalences. Further, we show that it is not possible to enlarge the class of weak equivalences to get cofibration category structure on the category of finite graphs without including morphisms where domain and codomain have non-isomorphic stiff subgraphs.