Spider Web Graphs

Abstract

Recent results from Chih-Scull show that the ×-homotopy relation on finite graphs can be expressed via a sequence of “spider moves” which shift a single vertex at a time. In this paper we study a “spider web graph” which encodes exactly these spider moves between graph homomorphisms. We show how composition of graph homomorphisms relate to the spider web, study the components of spider webs for bipartite and tree graphs, and
finish by giving an explicit description of the spider web for homomorphisms from a bipartite graph to a star graph.