Simplicial maps of graphs that factor through an arc

Abstract

In this paper it is shown that a simplicial map ϕ from a connected graph into a graph can be factored through an arc if and only if there are a monotone map μ, a sequence π 1, π 2,…, π n of folds, and an irreducible map ψ whose domain is an arc such that ϕ= ψ∘ π n ∘⋯∘ π 2∘ π 1∘ μ. Implicit in this result is a procedure that can be useful for determining whether a simplicial map factors through an arc. For those that do, the procedure produces a factorization. This is demonstrated by means of an example. On the way to the main results, quotient graphs are defined and fundamental results relating them to simplicial maps are proved. Folds are then defined as a specific type of projection onto a quotient graph, generalizing previous definitions.