Homotopy covers of graphs

Chih, Tien (Oxford College of Emory University)
Scull, Laura (Fort Lewis College)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 14, No 1 (2026): Electronic Journal of Graph Theory and Applications

We develop a theory of ×-homotopy, fundamental groupoids and covering spaces that applies to non-simple graphs, generalizing existing results for simple graphs. We prove that ×-homotopies from finite graphs can be decomposed into moves that adjust at most one vertex at a time, generalizing the spider lemma of Chih & Scull (2021). We define a notion of homotopy covering map and develop a theory of universal covers and deck transformations, generalizing Matsushita (2017) and Tardif–Wroncha (2019) to non-simple graphs. We examine the case of reflexive graphs (each vertex having at least one loop). We also prove that these homotopy covering maps satisfy a homotopy lifting property for arbitrary graph homomorphisms, generalizing path lifting results of Matsushita and Tardif–Wroncha.