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A graph with a universal vertex, u. 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 ...
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 ...
2. A universal vertex (also called an apex or dominating vertex) is a vertex that is adjacent to every other vertex in the graph. For instance, wheel graphs and connected threshold graphs always have a universal vertex. 3. In the logic of graphs, a vertex that is universally quantified in a formula may be called a universal vertex for that formula.
In graph theory, a wheel graph is a graph formed by connecting a single universal vertex to all vertices of a cycle. A wheel graph with n vertices can also be defined as the 1- skeleton of an ( n – 1 )-gonal pyramid .
However it is not the smallest such graph: it is known that there is a universal graph for n-vertex trees, with only n vertices and O(n log n) edges, and that this is optimal. [6] A construction based on the planar separator theorem can be used to show that n -vertex planar graphs have universal graphs with O( n 3/2 ) edges, and that bounded ...
A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
A universal traversal sequence is a sequence of instructions comprising a graph traversal for any regular graph with a set number of vertices and for any starting vertex. A probabilistic proof was used by Aleliunas et al. to show that there exists a universal traversal sequence with number of instructions proportional to O ( n 5 ) for any ...
An arc diagram. As well as for straight-line graph drawing, universal point sets have been studied for other drawing styles; in many of these cases, universal point sets with exactly n points exist, based on a topological book embedding in which the vertices are placed along a line in the plane and the edges are drawn as curves that cross this line at most once.