Geodetic topological cycles in locally finite graphs

We prove that the topological cycle space C(G) of a locally finite graph G is generated by its geodetic topological circles. We further show that, although the finite cycles of G generate C(G), its finite geodetic cycles need not generate C(G).Comment: 1

On the homology of locally finite graphs

We show that the topological cycle space of a locally finite graph is a canonical quotient of the first singular homology group of its Freudenthal compactification, and we characterize the graphs for which the two coincide. We construct a new singular-type homology for non-compact spaces with ends, which in dimension~1 captures precisely the topological cycle space of graphs but works in any dimension.Comment: 30 pages. This is an extended version of the paper "The homology of a locally finite graph with ends" (to appear in Combinatorica) by the same authors. It differs from that paper only in that it offers proofs for Lemmas 3, 4 and 10, as well as a new footnote in Section