Abrams's stable equivalence for graph braid groups

In his PhD thesis [1], Abrams proved that, for a natural number n and a graph G with at least n vertices, the n-strand configuration space of G, denoted C-n (C), deformation retracts to a compact subspace, the discretized n-strand configuration space, provided G satisfies two conditions: each path between distinct essential vertices (vertices of degree not equal to 2) is of length at least n + 1 edges, and each path from a vertex to itself which is not nullhomotopic is of length at least n + 1 edges. Using Forman's discrete Morse theory for CW-complexes, we show the first condition can be relaxed to require only that each path between distinct essential vertices is of length at least n - 1. (C) 2014 Elsevier B.V. All rights reserved.