cluster graph
{{Short description|Graph made from disjoint union of complete graphs}}
{{about|disjoint unions of complete graphs|graphs with clustered vertices|clustered planarity|visualizations of clusterings|cluster diagram}}
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 {{math|P{{sub|3}}}}-free graphs. They are the complement graphs of the complete multipartite graphs[http://www.graphclasses.org/classes/gc_1237.html Cluster graphs], Information System on Graph Classes and their Inclusions, accessed 2016-06-26. and the 2-leaf powers.{{citation
| last1 = Nishimura | first1 = N.
| last2 = Ragde | first2 = P.
| last3 = Thilikos | first3 = D.M.
| journal = Journal of Algorithms
| pages = 69–108
| title = On graph powers for leaf-labeled trees
| volume = 42
| year = 2002 | doi=10.1006/jagm.2001.1195}}. The cluster graphs are transitively closed, and every transitively closed undirected graph is a cluster graph.{{citation
| last1 = McColl | first1 = W. F.
| last2 = Noshita | first2 = K.
| doi = 10.1016/0166-218X(86)90020-X
| issue = 1
| journal = Discrete Applied Mathematics
| mr = 856101
| pages = 67–73
| title = On the number of edges in the transitive closure of a graph
| volume = 15
| year = 1986| doi-access = free
}}
The cluster graphs are the graphs for which adjacency is an equivalence relation, and their connected components are the equivalence classes for this relation.
Related graph classes
Every cluster graph is a block graph, a cograph, and a claw-free graph. Every maximal independent set in a cluster graph chooses a single vertex from each cluster, so the size of such a set always equals the number of clusters; because all maximal independent sets have the same size, cluster graphs are well-covered.
The Turán graphs are complement graphs of cluster graphs, with all complete subgraphs of equal or nearly-equal size. The locally clustered graph (graphs in which every neighborhood is a cluster graph) are the diamond-free graphs, another family of graphs that contains the cluster graphs.
When a cluster graph is formed from cliques that are all the same size, the overall graph is a homogeneous graph, meaning that every isomorphism between two of its induced subgraphs can be extended to an automorphism of the whole graph. With only two exceptions, the cluster graphs and their complements are the only finite homogeneous graphs,{{citation
| last = Gardiner | first = A.
| issue = 1
| journal = Journal of Combinatorial Theory
| mr = 0419293
| pages = 94–102
| series = Series B
| title = Homogeneous graphs
| volume = 20
| year = 1976
| doi=10.1016/0095-8956(76)90072-1| doi-access = free
}}. and infinite cluster graphs also form one of only a small number of different types of countably infinite homogeneous graphs.{{citation
| last1 = Lachlan | first1 = A. H.
| last2 = Woodrow | first2 = Robert E.
| doi = 10.2307/1999974
| issue = 1
| journal = Transactions of the American Mathematical Society
| mr = 583847
| pages = 51–94
| title = Countable ultrahomogeneous undirected graphs
| volume = 262
| year = 1980| doi-access = free
}}.
Computational problems
A subcoloring of a graph is a partition of its vertices into induced cluster graphs. Thus, the cluster graphs are exactly the graphs of subchromatic number 1.{{citation
| last1 = Albertson | first1 = M. O.
| last2 = Jamison | first2 = R. E.
| last3 = Hedetniemi | first3 = S. T.
| last4 = Locke | first4 = S. C.
| doi = 10.1016/0012-365X(89)90196-9
| issue = 1–2
| journal = Discrete Mathematics
| pages = 33–49
| title = The subchromatic number of a graph
| volume = 74
| year = 1989| doi-access = free
}}.
The computational problem of finding a small set of edges to add or remove from a graph to transform it into a cluster graph is called cluster editing. It is NP-complete{{citation
| last1 = Shamir | first1 = Ron
| last2 = Sharan | first2 = Roded
| last3 = Tsur | first3 = Dekel
| doi = 10.1016/j.dam.2004.01.007
| issue = 1-2
| journal = Discrete Applied Mathematics
| mr = 2095392
| pages = 173–182
| title = Cluster graph modification problems
| volume = 144
| year = 2004| doi-access = free
}}. but fixed-parameter tractable.{{citation
| last1 = Böcker | first1 = Sebastian
| last2 = Baumbach | first2 = Jan
| contribution = Cluster editing
| doi = 10.1007/978-3-642-39053-1_5
| mr = 3102002
| pages = 33–44
| publisher = Springer, Heidelberg
| series = Lecture Notes in Comput. Sci.
| title = The nature of computation
| volume = 7921
| year = 2013}}.
Given a complete graph with edge costs (positive and negative) the clique partitioning problem asks for a subgraph that is a cluster graph such that the sum of the costs of the edges of the cluster graph is minimal.{{citation
| last1 = Grötschel | first1 = Martin
| last2 = Wakabayashi | first2 = Yoshiko
| doi = 10.1007/BF01589097
| pages = 59–96
| publisher = Springer
| series = Mathematical Programming
| title = A cutting plane algorithm for a clustering problem
| volume = 45
| year = 1989}}.
This problem is closely related to the correlation clustering problem.