Simple homotopy of flag simplicial complexes and contractible contractions of graphs

Abstract

In his work on molecular spaces, Ivashchenko introduced the notion of an $I$-contractible transformation on a graph G, a family of addition/deletion operations on its vertices and edges. Chen, Yau, and Yeh used these operations to define the $I$-homotopy type of a graph, and showed that $I$-contractible transformations preserve the simple homotopy type of $C\left(G\right)$, the clique complex of G. In other work, Boulet, Fieux, and Jouve introduced the notion of s-homotopy of graphs to characterize the simple homotopy type of a flag simplicial complex. They proved that s-homotopy preserves $I$-homotopy, and asked whether the converse holds. In this note, we answer their question in the affirmative, concluding that graphs G and H are $I$-homotopy equivalent if and only if $C\left(G\right)$ and $C\left(H\right)$ are simple homotopy equivalent. We also show that a finite graph G is $I$-contractible if and only if $C\left(G\right)$ is contractible, which answers a question posed by the first author, Espinoza, Frías-Armenta, and Hernández. We use these ideas to give a characterization of simple homotopy for arbitrary simplicial complexes in terms of links of vertices.