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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 ...
Every graph is the disjoint union of its components. [2] Additional examples include the following special cases: In an empty graph, each vertex forms a component with one vertex and zero edges. [3] More generally, a component of this type is formed for every isolated vertex in any graph. [4]
Spectral graph theory is the branch of graph theory that uses spectra to analyze graphs. See also spectral expansion. split 1. A split graph is a graph whose vertices can be partitioned into a clique and an independent set. A related class of graphs, the double split graphs, are used in the proof of the strong perfect graph theorem.
A graph is said to be hyper-connected or hyper-κ if the deletion of each minimum vertex cut creates exactly two components, one of which is an isolated vertex. A graph is semi-hyper-connected or semi-hyper-κ if any minimum vertex cut separates the graph into exactly two components. [4]
A connected graph G with the same vertex set as a connected hypergraph H is a host graph for H if every hyperedge of H induces a connected subgraph in G. For a disconnected hypergraph H, G is a host graph if there is a bijection between the connected components of G and of H, such that each connected component G' of G is a host of the ...
If there are multiple maximum independent sets, only one need be output. This problem is sometimes referred to as "vertex packing". In the maximum-weight independent set problem, the input is an undirected graph with weights on its vertices and the output is an independent set with maximum total weight. The maximum independent set problem is ...
If at some point the remaining graph has an isolated vertex, the score is infinity; Otherwise, at some point the remaining graph contains no vertex; in that case the score is the number of explosions. For every given graph G, the game value on G (i.e., the score of CON when both sides play optimally) is denoted by Ψ(G).
In graph theory, a universal vertex is a vertex of an undirected graph that is adjacent to all other vertices of the graph. It may also be called a dominating vertex, as it forms a one-element dominating set in the graph. A graph that contains a universal vertex may be called a cone, and its universal vertex may be called the apex of the cone. [1]