Jordan circuits of a graph

AbstractLet G be connected, finite graph. Let C be a circuit of G. β(C), the strong bridge graph of C in G, is defined as follows: (1) the vertices of β(C) are the bridges of C in G, and (2) there is an edge in β(C) joining a pair of vertices B1 and B2 if and only if B1 and B2 separate each other relative C. •Theorem. Let G be a finite, connected graph. G is planar if and only if β(C) is bipartite for each circuit C in G.•Lemma. Let G be a finite, connected graph. G is not planar if and only if there is a circuit C of G for which β(C) contains a loop or a triangle.This Lemma isolates the crucial step in a new proof of the Kuratowski Theorem