A DECOMPOSITION OF LOCALLY FINITE GRAPHS

We prove that every infinite, connected, locally finite graph G can be expressed as an edge-disjoint union of a leafless tree T, rooted at a n arbitrarily chosen vertex of G, and a collection of finite graphs H- 1, H-2, H-3, ... such that, for all i less than j, the vertices common to H(i) and H(j) lie in T, and no vertex of H(j) lies on T between a vertex of H(i) and T and the root.