Intersection number (graph theory)

In the mathematical field of graph theory, the intersection number of a graph is the smallest number of elements in a representation of as an intersection graph of finite sets. In such a representation, each vertex is represented as a set, and two vertices are connected by an edge whenever their sets have a common element. Equivalently, the intersection number is the smallest number of cliques needed to cover all of the edges of . A set of cliques that cover all edges of a graph is called a clique edge cover or edge clique cover, or even just a clique cover, although the last term is ambiguous: a clique cover can also be a set of cliques that cover all vertices of a graph. Sometimes "covering" is used in place of "cover". As well as being called the intersection number, the minimum number of these cliques has been called the R-content, edge clique cover number, or clique cover number. The problem of computing the intersection number has been called the intersection number problem, the intersection graph basis problem, covering by cliques, the edge clique cover problem, and the keyword conflict problem. Every graph with vertices and edges has intersection number at most . The intersection number is NP-hard to compute or approximate, but fixed-parameter tractable. Let be any family of sets, allowing sets in to be repeated. Then the intersection graph of is an undirected graph that has a vertex for each set in and an edge between each two sets 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 such that there exists a representation of this type for which the union of the sets in has 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 is that it is the smallest number of cliques in (complete subgraphs of ) that together cover all of the edges of .