graph theory
graph
vertex, edge
omega-graph, hypergraph
quiver, n-quiver
reflexive, directed
relation
reflexive directed graph + unital associative composition = category
ribbon graph
In a graph an edge is an element that goes from one vertex to another.
If the graph underlies a category then an edge is a morphism.
If the graph underlies a simplicial set then an edge is a 1-dimensional simplex.