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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 ...
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-degree planar graphs have universal graphs with O(n log n) edges. [7] [8] [9] It is also possible to construct universal graphs for planar graphs that have n 1+o(1) vertices. [10]
A wheel graph with n vertices can also be defined as the 1-skeleton of an (n – 1)-gonal pyramid. Some authors [1] write W n to denote a wheel graph with n vertices (n ≥ 4); other authors [2] instead use W n to denote a wheel graph with n + 1 vertices (n ≥ 3), which is formed by connecting a single vertex to all vertices of a cycle of ...
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.
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).
The derived graph of the voltage graph has as its vertices the pairs (v,x) where v is a vertex of G and x is a group element; a dart from v to w labeled with the group element y in G corresponds to an edge from (v,x) to (w,xy) in the derived graph. The universal cover can be seen in this way as a derived graph of a voltage graph in which the ...
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 ...