In this chapter we introduce some basic definitions for graphs, maps, and polyhedra. We present here the basic notions. Further definitions will be introduced later when needed. The reader can consult the following books for more detailed information:.

Graphs

A graph G consists of a set V of vertices and a set E of edges such that each edge is assigned two vertices at its ends. Two vertices are adjacent if there is an edge between them. The degree of a vertex v ∈ V is the number of edges to which it is incident. A graph is said to be simple if no two edges have identical end-vertices, i.e. if it has no loops and multiple edges. In the special case of simple graphs, automorphisms are permutations of the vertices preserving adjacencies. For non-simple graphs (for example, when 2-gons occur) an automorphism of a graph is a permutation of the vertices and a permutation of the edges, preserving incidence between vertices and edges. By Aut(G) is denoted the group of automorphisms of the graph G; a synonym is symmetry group.

For U ⊆ V, let EU ⊆ E be the set of edges of a graph G = (V, E) having endvertices in U. Then the graph GU = (U, EU) is called the induced subgraph (by U) of G.

A graph G is said to be connected if, for any two of its vertices u, v, there is a path in G joining u and v.