Clique-Width is NP-Complete

@article{Fellows2009CliqueWidthIN,
  title={Clique-Width is NP-Complete},
  author={Michael R. Fellows and Frances A. Rosamond and Udi Rotics and Stefan Szeider},
  journal={SIAM J. Discret. Math.},
  year={2009},
  volume={23},
  pages={909-939}
}
Clique-width is a graph parameter that measures in a certain sense the complexity of a graph. Hard graph problems (e.g., problems expressible in monadic second-order logic with second-order quantification on vertex sets, which includes NP-hard problems such as 3-colorability) can be solved in polynomial time for graphs of bounded clique-width. We show that the clique-width of a given graph cannot be absolutely approximated in polynomial time unless $P = NP$. We also show that, given a graph $G… 
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101 Citations

Clique-Width for Hereditary Graph Classes
TLDR
A possible strong connection between results on boundedness of clique-width and on well-quasi-orderability by the induced subgraph relation for hereditary graph classes is explained.
A SAT Approach to Clique-Width
TLDR
This work presents a new method for computing the clique-width of graphs based on an encoding to propositional satisfiability (SAT) which is then evaluated by a SAT solver and determined the smallest graphs that require a small pre-described cliques-width.
Polynomial-time recognition of clique-width ≤3 graphs
On the Clique-Width of Unigraphs
TLDR
It is shown that every unigraph has clique-width at most 5.1, so it follows that many problems that are NP-hard in general are polynomial-time solvable for unigraphs.
Critical properties of graphs of bounded clique-width
A SAT Approach to Clique-Width
TLDR
This SAT-based method is the first to discover the exact clique-width of various small graphs, including famous named graphs from the literature as well as random graphs of various density.
Exploiting Restricted Linear Structure to Cope with the Hardness of Clique-Width
TLDR
A polynomial-time algorithm for a larger subclass of proper interval graphs that approximates the clique-width within an additive factor 3 is presented.
Computing the Clique-Width of Large Path Powers in Linear Time via a New Characterisation of Clique-Width
TLDR
This paper presents a new characterisation of clique-width based on rooted binary trees, completely without the use of labelled graphs, and a result that indicates that large k-path powers constitute the first non-trivial infinite class of graphs of unbounded cliques-width whoseClique- width can be computed exactly in polynomial time.
Tight Complexity Bounds for FPT Subgraph Problems Parameterized by Clique-Width
TLDR
The Dense (Sparse)k-Subgraph problem, which asks whether there exists an induced subgraph of G with k vertices and at least q edges (at most q edges, respectively), can be solved in time kO(w)·n, but it cannot be solve in time 2o(wlogk)·o(1) unless the Exponential Time Hypothesis (ETH) fails.
P-FPT algorithms for bounded clique-width graphs
TLDR
This paper studies several graph theoretic problems for which hardness results exist and gives hardness results and P-FPT algorithms, using clique-width and some of its upper bounds as parameters, based on preprocessing methods using modular decomposition and split decomposition.
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