vertexedge

Vertexedge is a term used in graph theory and related fields to describe the incidence relationship between a vertex and an edge in a graph. It is not a separate object on its own, but rather the pairing of a vertex with an edge to indicate that the vertex lies on that edge.

Formally, let G = (V, E) be a graph. The vertex-edge incidence relation I is a subset of V × E such that (v, e) ∈ I if and only if v is an endpoint of e. In an undirected graph where an edge e connects two vertices {u, w}, the incidences are (u, e) and (w, e). In a directed graph with an edge e = (u → w), there can be a distinction between tail-incidence (u, e) and head-incidence (w, e).

Representations of vertexedge include incidence structures such as incidence lists and incidence matrices. An incidence list stores, for each vertex, the set or list of incident edges, while an incidence matrix M has rows indexed by vertices and columns by edges, with M[v, e] = 1 if v is incident to e and 0 otherwise. These representations facilitate operations like computing vertex degrees, checking edge endpoints, and performing edge-centric traversals.

Applications of the vertexedge concept appear in algorithms that rely on incidences, such as edge-based traversals, network modeling, and circuit design. It also relates to more general incidence structures, and to the incidence graph of a graph, a bipartite graph that represents the original vertices and edges as separate node sets with incidences between them.

See also: graph theory, incidence structure, incidence matrix, adjacency list, incidence graph.