Intersection graph

An intersection graph has one vertex for each set or object in a collection, with two vertices adjacent exactly when their corresponding sets or objects intersect.

Graph (discrete mathematics): A structure consisting of vertices joined by edges. An intersection graph is a graph whose edges encode pairwise intersections.
Interval graph: An intersection graph of intervals on a line. Its edges arise exactly when two intervals overlap.
Scheduling: The assignment of tasks to times or resources subject to constraints. An interval graph models tasks that conflict because their time intervals overlap.
Comparability graph: A graph whose edges join pairs comparable in a partial order. Its adjacency comes from order rather than geometric or set intersection.
Set intersection: The operation that returns the elements common to two or more sets. Each edge records whether two represented sets have a nonempty intersection.
Chordal graph: A graph in which every cycle of length at least four has a chord. Intersection graphs of subtrees of a tree are chordal, linking overlap structure to cycle structure.
Register allocation: The compiler task of assigning program variables to a limited set of processor registers. Live ranges form an interference graph, an interval graph in a common simplified model.
Containment graph: A graph encoding containment relations among a family of sets or objects. Containment can define adjacency even when two objects do not intersect symmetrically.
Clique: A set of vertices in a graph that are pairwise adjacent. A clique corresponds to a collection of represented sets that intersect pairwise.
Perfect graph: A graph whose every induced subgraph has equal chromatic number and clique number. Many geometric intersection classes are perfect, making their coloring behavior tractable.

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Helly's theoremRelated: The theorem turns local intersection information into a global common intersection.