IntroductionSeveral characterisations of planar graphs are known. (See [1–20].) We present a new onebased on the structure of the cocycle space of a graph.Let G be a graph with vertex set V G and edge set EG. If S and T are disjoint setsof vertices, we denote by [S, T ] the set of edges with one end in S and the other in T .For any S ⊆ V G we write S = V G − S, and we define ∂S = [S, S]. This set is calleda cocycle, the cocycle determined by S (or S). A bond is a minimal non-empty cocycle.Thus a non-empty cocycle ∂S in a connected graph G is a bond if and only if G andG are both connected. An isthmus is the unique member of a bond of cardinality 1.Let A and B be distinct bonds in G. We say that two distinct edges of B − A are bound(to each other)...