Intersection number (graph theory)

In the mathematical field of graph theory, the intersection number of a graph G = (V,E) is the smallest number of elements in a representation of G as an intersection graph of finite sets. Equivalently, it is the smallest number of cliques needed to cover all of the edges of G.

Let F be a family of sets (allowing sets in F to be repeated); then the intersection graph of F is an undirected graph that has a vertex for each member of F and an edge between each two members that have a nonempty intersection. Every graph can be represented as an intersection graph in this way. The intersection number of the graph is the smallest number k such that there exists a representation of this type for which the union of F has k elements. The problem of finding an intersection representation of a graph with a given number of elements is known as the intersection graph basis problem.

An alternative definition of the intersection number of a graph G is that it is the smallest number of cliques in G (complete subgraphs of G) that together cover all of the edges of G. A set of cliques with this property is known as a clique edge cover or edge clique cover, and for this reason the intersection number is also sometimes called the edge clique cover number.

The equality of the intersection number and the edge clique cover number is straightforward to prove. In one direction, suppose that G is the intersection graph of a family F of sets whose union U has k elements. Then for any element x of U, the subset of vertices of G corresponding to sets that contain x forms a clique: any two vertices in this subset are adjacent, because their sets have a nonempty intersection containing x. Further, every edge in G is contained in one of these cliques, because an edge corresponds to a nonempty intersection and an intersection is nonempty if it contains at least one element of U. Therefore, the edges of G can be covered by k cliques, one per element of U. In the other direction, if a graph G can be covered by k cliques, then each vertex of G may be represented by the set of cliques that contain that vertex.

...
Wikipedia