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Figure 1: A faithful projection on R2 of a non-planar graph. Highlighted is a loop whose projectionexhibits a non-vertex edge crossing, i.e., one which is not realized in the original graph.Such crossings are excluded in the planar projections which are discussed in Theorem 4.3.Their effects is presented in Theorem 4.4.
Figure 2: The different resolutions of a vertex of order 4, which together determine the admissibleloop decompositions of an even subgraph Γ
Figure 3: An even number of lines meeting at a vertex, and one of their possible pairings, in thiscase of parity (−1)3 = −1.
Figure 4: Order-disorder variables for a planar graph. The disorder variables τℓj are defined througha set of lines ℓj, each linking a dual site x∗j ∈ G∗ which is a neighbor of xj in G × G∗with a common dual site x∗0 ∈ G∗, called grand central
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