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Vertex identification (sometimes called vertex contraction) removes the restriction that the contraction must occur over vertices sharing an incident edge. (Thus, edge contraction is a special case of vertex identification.) The operation may occur on any pair (or subset) of vertices in the graph.
In graph theory, a vertex is incident with an edge if the vertex is one of the two vertices the edge connects. An incidence is a pair ( u , e ) {\displaystyle (u,e)} where u {\displaystyle u} is a vertex and e {\displaystyle e} is an edge incident with u {\displaystyle u} .
A graph with 6 vertices and 7 edges where the vertex number 6 on the far-left is a leaf vertex or a pendant vertex. In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph ...
A vertex cover is a set of vertices incident to every edge in a graph. An edge cover is a set of edges incident to every vertex in a graph. A set of subgraphs of a graph covers that graph if its union – taken vertex-wise and edge-wise – is equal to the graph. critical
Given a graph G, we denote the set of its edges by E(G) and that of its vertices by V(G). Let q be the cardinality of E(G) and p be that of V(G). Once a labeling of the edges is given, a vertex of the graph is labeled by the sum of the labels of the edges incident to it, modulo p. Or, in symbols, the induced labeling on a vertex is given by
An edge-graceful labeling on a simple graph without loops or multiple edges on p vertices and q edges is a labeling of the edges by distinct integers in {1, …, q} such that the labeling on the vertices induced by labeling a vertex with the sum of the incident edges taken modulo p assigns all values from 0 to p − 1 to the vertices.
The degree or valency of a vertex is the number of edges that are incident to it; for graphs with loops, a loop is counted twice. In a graph of order n, the maximum degree of each vertex is n − 1 (or n + 1 if loops are allowed, because a loop contributes 2 to the degree), and the maximum number of edges is n(n − 1)/2 (or n(n + 1)/2 if loops ...
The link of a vertex of a tetrahedron is the triangle. An alternative definition is: the link of a vertex () is the graph Lk(v, X) constructed as follows. The vertices of Lk(v, X) are the edges of X incident to v. Two such edges are adjacent in Lk(v, X) iff they are incident to a common 2-cell at v.