In graph theory, the clique-width of a graph is a parameter that describes the structural complexity of the graph; it is closely related to treewidth, but unlike treewidth it can be small for dense graphs.It is defined as the minimum number of labels needed to construct by means of the following 4 operations :
G ⊕ H
Graphs of bounded clique-width include the cographs and distance-hereditary graphs. Although it is NP-hard to compute the clique-width when it is unbounded, and unknown whether it can be computed in polynomial time when it is bounded, efficient approximation algorithms for clique-width are known.Based on these algorithms and on Courcelle's theorem, many graph optimization problems that are NP-hard for arbitrary graphs can be solved or approximated quickly on the graphs of bounded clique-width.
The construction sequences underlying the concept of clique-width were formulated by Courcelle, Engelfriet, and Rozenberg in 1990 and by . The name "clique-width" was used for a different concept by . By 1993, the term already had its present meaning.
Cographs are exactly the graphs with clique-width at most 2. Every distance-hereditary graph has clique-width at most 3. However, the clique-width of unit interval graphs is unbounded (based on their grid structure). Similarly, the clique-width of bipartite permutation graphs is unbounded (based on similar grid structure). Based on the characterization of cographs as the graphs with no induced subgraph isomorphic to a path with four vertices, the clique-width of many graph classes defined by forbidden induced subgraphs has been classified.[1]
Other graphs with bounded clique-width include the -leaf powers for bounded values of ; these are the induced subgraphs of the leaves of a tree in the graph power . However, leaf powers with unbounded exponents do not have bounded clique-width.[2]
and proved the following bounds on the clique-width of certain graphs:
Additionally, if a graph has clique-width, then the graph power has clique-width at most . Although there is an exponential gap in both the bound for clique-width from treewidth and the bound for clique-width of graph powers, these bounds do not compound each other:if a graph has treewidth, then has clique-width at most, only singly exponential in the treewidth.
Many optimization problems that are NP-hard for more general classes of graphs may be solved efficiently by dynamic programming on graphs of bounded clique-width, when a construction sequence for these graphs is known. In particular, every graph property that can be expressed in MSO1 monadic second-order logic (a form of logic allowing quantification over sets of vertices) has a linear-time algorithm for graphs of bounded clique-width, by a form of Courcelle's theorem.
It is also possible to find optimal graph colorings or Hamiltonian cycles for graphs of bounded clique-width in polynomial time, when a construction sequence is known, but the exponent of the polynomial increases with the clique-width, and evidence from computational complexity theory shows that this dependence is likely to be necessary.The graphs of bounded clique-width are -bounded, meaning that their chromatic number is at most a function of the size of their largest clique.
The graphs of clique-width three can be recognized, and a construction sequence found for them, in polynomial time using an algorithm based on split decomposition.For graphs of unbounded clique-width, it is NP-hard to compute the clique-width exactly, and also NP-hard to obtain an approximation with sublinear additive error. However, when the clique-width is bounded, it is possible to obtain a construction sequence of bounded width (exponentially larger than the actual clique-width) in polynomial time,[6] in particular in quadratic time in the number of vertices. It remains open whether the exact clique-width, or a tighter approximation to it, can be calculated in fixed-parameter tractable time, whether it can be calculated in polynomial time for every fixed bound on the clique-width, or even whether the graphs of clique-width four can be recognized in polynomial time.
The theory of graphs of bounded clique-width resembles that for graphs of bounded treewidth, but unlike treewidth allows for dense graphs. If a family of graphs has bounded clique-width, then either it has bounded treewidth or every complete bipartite graph is a subgraph of a graph in the family. Treewidth and clique-width are also connected through the theory of line graphs: a family of graphs has bounded treewidth if and only if their line graphs have bounded clique-width.
The graphs of bounded clique-width also have bounded twin-width.