(Ordinary) Graph

A graph g with the vertices V -or nodes- and edges E is denoted as g = $\langle$G,V$\rangle$. The set of edges E is a set of ordinary pairs like E = {{v,v'}$\mid$ v, v' $in$ V }. A graph is called finite if the cardinality of the set of vertices V as well as the set of edges E is countable. A graph is called infinite if either the set of vertices V or the set of edges E is not countable, i.e. either the set of vertices or the set of edges each have infinite cardinality. A $path \pi$ in a graph G is a sequence of vertices $\pi = v_{0}, v_{1}, ..., v_{i}...$ with i $\geq$ 0 and such that for each i with $1 \leq i$ there is an edge $(v_{i}, v_{i+1}) \in E$. A path is called a cycle if $v_{0} = v_{i}$ for some $i \geq 1$.