Week 5 Ribbon graphs, duals and embeddings

Question 1

Graph - embedding version

Answer 1

A graph is a quadruple (V,H,τ,ω) consisting of two sets, V (vertices) and H (half-edges), a fixed-point free involution
τ: H →H and a function ω: H →V.

Question 2

Ribbon graph

Answer 2

A ribbon graph is a graph together with a cyclic ordering of
the half-edges around each vertex.

Question 3

Genus of a graph

Answer 3

Let Γ be a ribbon graph and let v, e and f denote the
number of vertices, edges and faces respectively. We define
the genus g of the ribbon graph Γ by the formula
v−e+ f = 2−2g.

Question 4

Dual ribbon graph

Answer 4

The dual ribbon graph to Γ, denoted Γ∗ has vertex set
V∗= F, half-edge set H∗ (= E x {+, -}) and the maps τ∗ : H∗ →H∗ and ω∗ : H∗ →V∗ are defined by
τ((e,±)) = (e,∓),
ω((e,±)) = f ,