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 .[1][2]
A set of cliques that cover all edges of a graph is called a clique edge cover[3] or edge clique cover,[4] 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.[5] Sometimes "covering" is used in place of "cover".[6] As well as being called the intersection number, the minimum number of these cliques has been called the R-content,[7] edge clique cover number,[4] or clique cover number.[8] The problem of computing the intersection number has been called the intersection number problem,[9] the intersection graph basis problem,[10] covering by cliques,[10] the edge clique cover problem,[9] and the keyword conflict problem.[2]
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.
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