The main subjects of study of this book are the graphs, embedded in surfaces (2D manifolds), their local and global properties, and connections between them.

A graph *G = {V, E}* is a pair of sets, where *E\in\P(V)* is a subset of the power set *P(V)*. We will refer to the elements of *V* as vertices, and elements of *E* as edges or hyper-edges.

A *2D* manifold is a topological space, such that around every point *p* of the manifold there is an open set *U* homeomorphic to the unit disc *D*, and the homeomorphisms are coherent.

We will say that a graph *G* is embedded in a manifold *M*, if vertices ~ points, curves ~edges... In topological graph theory, an **embedding** (also spelled **imbedding**) of a graph on a surface Σ is a representation of on Σ in which points of Σ are associated to vertices and simple arcs (homeomorphic images of [0,1]) are associated to edges in such a way that:

- the endpoints of the arc associated to an edge are the points associated to the end vertices of ,
- no arcs include points associated with other vertices,
- two arcs never intersect at a point which is interior to either of the arcs.

A directed graph with boundary is a triple G = (V,\partial V,E}, where V is a set of points in a space, called *vertices*, \partial V is the subset of V called *boundary vertices*, and E is a set of directed pairs of elements of V, called *edges*.

- For every edge
*(a,b)*in*E*we call*b*a neighbor of*a*.