Clique-width



In graph theory, the clique-width of a graph $G$ 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 :


 * 1) Creation of a new vertex $v$ with label $i$ (denoted by $i(v)$)
 * 2) Disjoint union of two labeled graphs $G$ and $H$ (denoted by $$G \oplus H$$)
 * 3) Joining by an edge every vertex labeled $i$ to every vertex labeled $j$ (denoted by $η(i,j)$), where $i ≠ j$
 * 4) 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 and by. The name "clique-width" was used for a different concept by. By 1993, the term already had its present meaning.

Special classes of graphs
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.

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 power $T^{k}$. However, leaf powers with unbounded exponents do not have bounded clique-width.

Bounds
and proved the following bounds on the clique-width of certain graphs:
 * If a graph has clique-width at most $k$, then so does every induced subgraph of the graph.
 * The complement graph of a graph of clique-width $k$ has clique-width at most $2k$.
 * The graphs of treewidth $w$ have clique-width at most $3 &middot; 2^{w &minus; 1}$. The exponential dependence in this bound is necessary: there exist graphs whose clique-width is exponentially larger than their treewidth. 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 $n &minus; 1$. However, graphs of clique-width $k$ that have no complete bipartite graph $K_{t,t}$ as a subgraph have treewidth at most $3k(t &minus; 1) &minus; 1$. Therefore, for every family of sparse graphs, having bounded treewidth is equivalent to having bounded clique-width.
 * Another graph parameter, the rank-width, is bounded in both directions by the clique-width: rank-width ≤ clique-width ≤ 2rank-width + 1.

Additionally, if a graph $G$ has clique-width $k$, then the graph power $G^{c}$ has clique-width at most $2kc^{k}$. 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 $G^{c}$ has clique-width at most $2(c + 1)^{w + 1} &minus; 2$, only singly exponential in the treewidth.

Computational complexity
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 $χ$-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, 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.

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. 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.