 Intersection Graph
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Intersection Graph

In graph theory, an intersection graph is a graph that represents the pattern of intersections of a family of sets. Any graph can be represented as an intersection graph, but some important special classes of graphs can be defined by the types of sets that are used to form an intersection representation of them.

## Formal definition

Formally, an intersection graph G is an undirected graph formed from a family of sets

Si, i = 0, 1, 2, ...

by creating one vertex vi for each set Si, and connecting two vertices vi and vj by an edge whenever the corresponding two sets have a nonempty intersection, that is,

E(G) = {{vivj} | Si ? Sj ? ?}.

## All graphs are intersection graphs

Any undirected graph G may be represented as an intersection graph: for each vertex vi of G, form a set Si consisting of the edges incident to vi; then two such sets have a nonempty intersection if and only if the corresponding vertices share an edge. Erd?s, Goodman & Pósa (1966) provide a construction that is more efficient (which is to say requires a smaller total number of elements in all of the sets Si combined) in which the total number of set elements is at most n2/4 where n is the number of vertices in the graph. They credit the observation that all graphs are intersection graphs to Szpilrajn-Marczewski (1945), but say to see also ?ulík (1964). The intersection number of a graph is the minimum total number of elements in any intersection representation of the graph.

## Classes of intersection graphs

Many important graph families can be described as intersection graphs of more restricted types of set families, for instance sets derived from some kind of geometric configuration:

Scheinerman (1985) characterized the intersection classes of graphs, families of finite graphs that can be described as the intersection graphs of sets drawn from a given family of sets. It is necessary and sufficient that the family have the following properties:

• Every induced subgraph of a graph in the family must also be in the family.
• Every graph formed from a graph in the family by replacing a vertex by a clique must also belong to the family.
• There exists an infinite sequence of graphs in the family, each of which is an induced subgraph of the next graph in the sequence, with the property that every graph in the family is an induced subgraph of a graph in the sequence.

If the intersection graph representations have the additional requirement that different vertices must be represented by different sets, then the clique expansion property can be omitted.

## Related concepts

An order-theoretic analog to the intersection graphs are the inclusion orders. In the same way that an intersection representation of a graph labels every vertex with a set so that vertices are adjacent if and only if their sets have nonempty intersection, so an inclusion representation f of a poset labels every element with a set so that for any x and y in the poset, x y if and only if f(x) ? f(y).