Graphs on Surfaces by Joanna A. Ellis-Monaghan & Iain Moffatt

Author:Joanna A. Ellis-Monaghan & Iain Moffatt
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

Figure 4.1 shows the local effect of contracting an edge of a ribbon graph.

Fig. 4.1Contracting an edge of a ribbon graph

We note that edge contraction as defined in Definition 4.4 was considered by Bollobás and Riordan in Sect. 7 of [6] (where the language of partial duals was not used) and was observed to agree with G δ(e) − e by Chmutov in [16].

We conclude by setting up some additional notation. If G is an embedded graph and A ⊆ E(G), then G − A is the embedded graph obtained from G by deleting all of the edges in A and G ∕ A is the embedded graph obtained by contracting all of the edges in A.

Deletion and contraction are reflected in the medial graph. For Proposition 4.5 we use the notation for vertex states from Sect. 1.5.2.

Proposition 4.5.

Let G be an embedded graph with embedded, canonically checkerboard coloured medial graph G m , and let e be any edge of G, with v e the associated vertex in G m . Then 1. .

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