2023/12/18 by Dochtermann, Anton, Matsushita, Takahiro
#04E45 #57Q10 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2312.10976
In his work on molecular spaces, Ivashchenko introduced the notion of an \mathfrakI-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 \mathfrakI-homotopy type of a graph, and showed that \mathfrakI-contractible transformations preserve the simple homotopy type of C(G), 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 \mathfrakI-homotopy, and asked whether the converse holds. In this note, we answer their question in the affirmative, concluding that graphs G and H are \mathfrakI-homotopy equivalent if and only if C(G) and C(H) are simple homotopy equivalent. We also show that a finite graph G is \mathfrakI-contractible if and only if C(G) 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.