A directed graph is a set of nodes (or vertices) and a set of edges (disjoint from the nodes), where each edge has an initial node and a terminal node. 

More formally, a directed graph is a quadruple (V, E, α, β), where

• V is a (possibly infinite) set of nodes,
• E is a set of edges, disjoint from V,
• α and β are two mappings from each edge e ∈ E, with α(e) ∈ V the origin (or initial vertex) of e and β(e) ∈ V the terminal (or end vertex) of e.  α(e) and β(e) are the endpoints of e. 

An undirected graph is a set of nodes (or vertices) and a set of edges (disjoint from the nodes), where each edge has two endpoints, not distinguished from each other. 

More formally, an undirected graph is a triple (V, E, δ), where

• δ is a mapping from each edge e ∈ E to two nodes (not necessarily distinct) v1 ∈ V and v2 ∈ V. v1 and v2 are the endpoints of e. 

Two directed graphs (V, E, α, β) and (V', E', α', β') are isomorphic if there are bijections g:V→V' and h:E→E' such that for each e∈E,

A loop is an edge both whose endpoints are equal. 

A graph contains multiple edges when two edges share corresponding endpoints.  If a directed graph has no multiple edges, then for it E⊆V×V.

A graph is simple if it contains neither loops nor multiple edges. 

Informally, a partial graph is obtained by deleting some edges, and a subgraph is obtained by deleting some nodes and any edges that are their endpoints.

G'=(V', E', α', β') is a partial graph of G=(V, E, α, β) if

• V'=V,
• E'⊆E, and
• ∀e∈E' : α'(e) = α(e) and β'(e) = β(e)

G' is a subgraph of G if

• V'⊆V,
• E' = {e∈E | α(e)∈V' and β(e)∈V' }, and
• ∀e∈E' : α'(e) = α(e) and β'(e) = β(e)