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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.
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 ...
The degree or valency of a vertex is the number of edges that are incident to it, where a loop is counted twice. The degree of a graph is the maximum of the degrees of its vertices. In an undirected simple graph of order n , the maximum degree of each vertex is n − 1 and the maximum size of the graph is n ( n − 1) / 2 .
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
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} .
An embedded graph uniquely defines cyclic orders of edges incident to the same vertex. The set of all these cyclic orders is called a rotation system.Embeddings with the same rotation system are considered to be equivalent and the corresponding equivalence class of embeddings is called combinatorial embedding (as opposed to the term topological embedding, which refers to the previous ...
Cromwell (1999) forms the vertex figure by intersecting the polyhedron with a sphere centered at the vertex, small enough that it intersects only edges and faces incident to the vertex. This can be visualized as making a spherical cut or scoop, centered on the vertex. The cut surface or vertex figure is thus a spherical polygon marked on this ...