Universal vertex

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] This terminology should be distinguished from the unrelated usage of these words for universal quantifiers in the logic of graphs, and for apex graphs.

This, in turn, can be used to show that the property of having a universal graph is evasive: testing this property may require checking the adjacency of all pairs of vertices.

However, a universal vertex can be recognized immediately from its degree: in an

Universal vertices can be described by a short logical formula, which has been used in graph algorithms for related properties.

More strongly they may be characterized as the finite graphs in which every connected induced subgraph contains a universal vertex.

They may be defined as the graphs that can be formed by repeated addition of either a universal vertex or an isolated vertex (one with no incident edges).

[4] In geometry, the three-dimensional pyramids have wheel graphs as their skeletons,[5] and more generally a higher-dimensional pyramid is a polytope whose faces of all dimensions connect an apex vertex to all the faces of a lower-dimensional base, including all of the vertices of the base.

[6] The friendship theorem states that, if every two vertices in a finite graph have exactly one shared neighbor, then the graph contains a universal vertex.

The graphs described by this theorem are the friendship graphs, formed by systems of triangles connected together at a common shared vertex, the universal vertex.

[7] The assumption that the graph is finite is important; there exist infinite graphs in which every two vertices have one shared neighbor, but with no universal vertex.

In a graph with a universal vertex, any removal sequence that leaves the universal vertex in place, removing all of the other vertices, fits this definition.

Almost all dismantlable graphs have a universal vertex, in the sense that the fraction of

-vertex dismantlable graphs that have a universal vertex tends to one in the limit as

This implies that a strong product has a dominating vertex if and only if both of its factors do; in this case the upper bound on its dominating number is one, and in any other case the lower bound is greater than one.

vertices, at least one of which is universal (or equivalently isolated, in the complement graph) can be counted by the inclusion–exclusion principle, in which one counts the graphs in which one chosen vertex is universal, then corrects for overcounting by subtracting the counts for graphs with two chosen universal vertices, then adding the counts for graphs with three chosen universal vertices, etc.

is the number of pairs of vertices that do not include a chosen universal vertex, and taking this number as the exponent of a power of two counts the number of graphs with the chosen vertices as universal.

[12] The unlabeled version of this graph enumeration problem is trivial, in the sense that the number of

-vertex unlabeled graphs with a universal vertex is the same as the number of

[10] The property of having a universal vertex can be expressed by a formula in the first-order logic of graphs.

The existence of this formula, and its small number of alternations between universal and existential quantifers, can be used in a fixed-parameter tractable algorithm for testing whether all components of a graph can be made to have universal vertices by

[14] The property of having a universal vertex (or equivalently an isolated vertex) has been considered with respect to the Aanderaa–Karp–Rosenberg conjecture on how many queries (subroutine calls) are needed to test whether a labeled graph has a property, given access to the graph only through a subroutine that can test whether two given vertices are adjacent.

vertices, one can determine the entire graph, and test any property, using

Testing the existence of a universal vertex is evasive, on graphs with an even number of vertices.

There are an odd number of these graphs that have a universal vertex.

A testing algorithm can be forced to query all pairs of vertices by an adjacency subroutine that always answers in such a way as to leave an odd number of remaining graphs that have a universal vertex.

Until all edges are tested, the total number of remaining graphs will be even, so the algorithm will be unable to determine whether the graph it is querying has a universal vertex.