Cluster graph

In graph theory, a branch of mathematics, a cluster graph is a graph formed from the disjoint union of complete graphs. Equivalently, a graph is a cluster graph if and only if it has no three-vertex induced path; for this reason, the cluster graphs are also called P3-free graphs. They are the complement graphs of the complete multipartite graphs and the 2-leaf powers. The cluster graphs are transitively closed, and every transitively closed undirected graph is a cluster graph.

This article is about disjoint unions of complete graphs. For graphs with clustered vertices, see clustered planarity. For visualizations of clusterings, see cluster diagram.

The cluster graphs are the graphs for which adjacency is an equivalence relation, and their connected components are the equivalence classes for this relation.

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