Universal vertex

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. (It is not to be confused with a universally quantified vertex in the logic of graphs.)

A graph with a universal vertex, u

A graph that contains a universal vertex may be called a cone. In this context, the universal vertex may also be called the apex of the cone. However, this terminology conflicts with the terminology of apex graphs, in which an apex is a vertex whose removal leaves a planar subgraph.

In special families of graphs

The stars are exactly the trees that have a universal vertex, and may be constructed by adding a universal vertex to an independent set. The wheel graphs, similarly, may be formed by adding a universal vertex to a cycle graph. The trivially perfect graphs (the comparability graphs of order-theoretic trees) always contain a universal vertex, the root of the tree, and more strongly they may be characterized as the graphs in which every connected induced subgraph contains a universal vertex. The connected threshold graphs form a subclass of the trivially perfect graphs, so they also contain 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).

In geometry, the three-dimensional pyramids have wheel graphs as their skeletons, 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. The polytope is said to be pyramidal at its apex, and it may have more than one apex. However, the existence of neighborly polytopes means that the graph of a polytope may have a universal vertex, or all vertices universal, without the polytope itself being a pyramid.

The friendship theorem of Paul Erdős, Alfréd Rényi, and Vera T. Sós (1966) 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. 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.

Every graph with a universal vertex is a dismantlable graph, meaning that it can be reduced to a single vertex by repeatedly removing vertices whose closed neighborhoods are subsets of other vertices' closed neighborhoods. 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 $n$-vertex dismantlable graphs that have a universal vertex tends to one in the limit as $n$ goes to infinity.

When a graph has a universal vertex, the vertex set consisting only of that vertex is a dominating set, a set that includes or is adjacent to every vertex. For this reason, in the context of dominating set problems, a universal vertex may also be called a dominating vertex. For the strong product of graphs $G\boxtimes H$, the domination numbers $\gamma (G)$ and $\gamma (H)$ obey the inequalities

$$\max\{\gamma (G),\gamma (H)\}\leq \gamma (G\boxtimes H)\leq \gamma (G)\gamma (H).$$

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.

Combinatorial enumeration

The number of labeled graphs with $n$ 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. This produces the formula

$\displaystyle \sum _{k=1}^{n}(-1)^{k+1}{\binom {n}{k}}2^{\tbinom {n-k}{2}}.$

In each term of the sum, $k$ is the number of vertices chosen to be universal, and ${\tbinom {n}{k}}$ is the number of ways to make this choice. ${\tbinom {n-k}{2}}$ 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.

Starting from $n=1$, these numbers of graphs are:

1, 1, 4, 23, 256, 5319, 209868, 15912975, 2343052576, 675360194287, ... (sequence A327367 in the OEIS).

For $n>1$, these numbers are odd when $n$ is even, and vice versa.

Recognition

In a graph with $n$ vertices, a universal vertex is a vertex whose degree is exactly $n-1$.

The property of having a universal vertex can be expressed by a formula in the first-order logic of graphs. Using $\sim$ to indicate the adjacency relation in a graph, a graph $G$ has a universal vertex if and only if it models the formula

$$\exists u\forall v{\bigl (}(u\neq v)\Rightarrow (u\sim v){\bigr )}.$$

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 $k$ steps of removing a vertex from each component.

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. In a graph with $n$ vertices, one can determine the entire graph, and test any property, using ${\tbinom {n}{2}}$ queries. A graph property is evasive if no algorithm can test the property guaranteeing fewer queries. 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.