In graph theory, the clique-width of a graphG 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 G by means of the following 4 operations :
Creation of a new vertex v with label i (denoted by i(v))
Disjoint union of two labeled graphs G and H (denoted by )
Joining by an edge every vertex labeled i to every vertex labeled j (denoted by η(i,j)), where i ≠ j
Renaming label i to label j (denoted by ρ(i,j))
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[1] and by Wanke (1994). The name "clique-width" was used for a different concept by Chlebíková (1992). By 1993, the term already had its present meaning.[2]
Special classes of graphs
Cographs are exactly the graphs with clique-width at most 2.[3]
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).[4]
Similarly, the clique-width of bipartite permutation graphs is unbounded (based on similar grid structure).[5]
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.[6]
Other graphs with bounded clique-width include the k-leaf powers for bounded values of k; these are the induced subgraphs of the leaves of a tree T in the graph powerTk. However, leaf powers with unbounded exponents do not have bounded clique-width.[7]
If a graph has clique-width at most k, then so does every induced subgraph of the graph.[8]
The complement graph of a graph of clique-width k has clique-width at most 2k.[9]
The graphs of treewidthw have clique-width at most 3 · 2w − 1. The exponential dependence in this bound is necessary: there exist graphs whose clique-width is exponentially larger than their treewidth.[10] In the other direction, graphs of bounded clique-width can have unbounded treewidth; for instance, n-vertex complete graphs have clique-width 2 but treewidth n − 1. However, graphs of clique-width k that have no complete bipartite graph Kt,t as a subgraph have treewidth at most 3k(t − 1) − 1. Therefore, for every family of sparse graphs, having bounded treewidth is equivalent to having bounded clique-width.[11]
Another graph parameter, the rank-width, is bounded in both directions by the clique-width: rank-width ≤ clique-width ≤ 2rank-width + 1.[12]
Additionally, if a graph G has clique-width k, then the graph powerGc has clique-width at most 2kck.[13] 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 G has treewidth w, then Gc has clique-width at most 2(c + 1)w + 1 − 2, only singly exponential in the treewidth.[14]
Computational complexity
Unsolved problem in mathematics:
Can graphs of bounded clique-width be recognized in polynomial time?
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.[15][16] In particular, every graph property that can be expressed in MSO1monadic 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.[16]
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.[17]
The graphs of bounded clique-width are χ-bounded, meaning that their chromatic number is at most a function of the size of their largest clique.[18]
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.[19]
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.[20] 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,[21] in particular in quadratic time in the number of vertices.[22] 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.[20]
Related width parameters
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.[11] 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.[23]
The graphs of bounded clique-width also have bounded twin-width.[24]
Brandstädt, Andreas; Hundt, Christian (2008), "Ptolemaic graphs and interval graphs are leaf powers", LATIN 2008: Theoretical informatics, Lecture Notes in Comput. Sci., vol. 4957, Springer, Berlin, pp. 479–491, doi:10.1007/978-3-540-78773-0_42, MR2472761.
Courcelle, Bruno; Engelfriet, Joost; Rozenberg, Grzegorz (1993), "Handle-rewriting hypergraph grammars", Journal of Computer and System Sciences, 46 (2): 218–270, doi:10.1016/0022-0000(93)90004-G, MR1217156. Presented in preliminary form in Graph grammars and their application to computer science (Bremen, 1990), MR1431281.
Courcelle, B. (1993), "Monadic second-order logic and hypergraph orientation", Proceedings of Eighth Annual IEEE Symposium on Logic in Computer Science (LICS '93), pp. 179–190, doi:10.1109/LICS.1993.287589, S2CID39254668.
Dvořák, Zdeněk; Král', Daniel (2012), "Classes of graphs with small rank decompositions are χ-bounded", Electronic Journal of Combinatorics, 33 (4): 679–683, arXiv:1107.2161, doi:10.1016/j.ejc.2011.12.005, S2CID5530520
Fomin, Fedor V.; Korhonen, Tuukka (2022), "Fast FPT-approximation of branchwidth", Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, ACM, pp. 886–899, arXiv:2111.03492, doi:10.1145/3519935.3519996, S2CID243832882.
Gurski, Frank; Wanke, Egon (2000), "The tree-width of clique-width bounded graphs without Kn,n", in Brandes, Ulrik; Wagner, Dorothea (eds.), Graph-Theoretic Concepts in Computer Science: 26th International Workshop, WG 2000, Konstanz, Germany, June 15–17, 2000, Proceedings, Lecture Notes in Computer Science, vol. 1928, Berlin: Springer, pp. 196–205, doi:10.1007/3-540-40064-8_19, MR1850348.
Todinca, Ioan (2003), "Coloring powers of graphs of bounded clique-width", Graph-theoretic concepts in computer science, Lecture Notes in Comput. Sci., vol. 2880, Springer, Berlin, pp. 370–382, doi:10.1007/978-3-540-39890-5_32, MR2080095.