Sphericity (graph theory)

In graph theory, the sphericity of a graph is a graph invariant defined to be the smallest dimension of Euclidean space required to realize the graph as an intersection graph of unit spheres. The sphericity of a graph is a generalization of the boxicity and cubicity invariants defined by F.S. Roberts in the late 1960s.^[1][2] The concept of sphericity was first introduced by Hiroshi Maehara in the early 1980s.^[3]

A graph of the vertices of a pentagon, realized as an intersection graph of disks in the plane. This is an example of a graph with sphericity 2, also known as a unit disk graph.

Definition

Let ${\displaystyle G}$ be a graph. Then the sphericity of ${\displaystyle G}$, denoted by ${\displaystyle \operatorname {sph} (G)}$, is the smallest integer ${\displaystyle n}$ such that ${\displaystyle G}$ can be realized as an intersection graph of unit spheres in ${\displaystyle n}$-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}$.^[4]

Sphericity can also be defined using the language of space graphs as follows. For a finite set of points in some ${\displaystyle n}$-dimensional Euclidean space, a space graph is built by connecting pairs of points with a line segment when their Euclidean distance is less than some specified constant. Then the sphericity of a graph ${\displaystyle G}$ is the minimum ${\displaystyle n}$ such that ${\displaystyle G}$ is isomorphic to a space graph in ${\displaystyle \mathbb {R} ^{n}}$.^[3]

Graphs of sphericity 1 are known as interval graphs or indifference graphs. Graphs of sphericity 2 are known as unit disk graphs.

Bounds

The sphericity of certain graph classes can be computed exactly. The following sphericities were given by Maehara on page 56 of his original paper on the topic.

Graph	Description	Sphericity	Notes
${\displaystyle K_{1}}$	Complete graph	0
${\displaystyle K_{n}}$	Complete graph	1	${\displaystyle n>1}$
${\displaystyle P_{n}}$	Path graph	1	${\displaystyle n>1}$
${\displaystyle C_{n}}$	Circuit graph	2	${\displaystyle n>3}$
${\displaystyle K_{m(2)}}$	Complete m-partite graph on m sets of size 2	2	${\displaystyle m>1}$

The most general known upper bound on sphericity is as follows. Assuming the graph is not complete, then ${\displaystyle \operatorname {sph} (G)\leq |G|-\omega (G)}$ where ${\displaystyle \omega (G)}$ is the clique number of ${\displaystyle G}$ and ${\displaystyle |G|}$ denotes the number of vertices of ${\displaystyle G.}$^[3]