Clique-Width is NP-Complete
SIAM Journal on Discrete Mathematics (SIDMA) vol. 23, no. 2, pp. 909-939, 2009.
A preliminary and shortened version of this paper appeared in the proceedings of STOC 2006; 38th ACM Symposium on Theory of Computing, Seattle, Washington, USA, pp. 354—362, ACM Press, 2006.
This paper combines the results of the technical reports:
Abstract: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, that includes NP-hard problems) can be solved efficiently for graphs of certified small 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 and an integer k, deciding whether the clique-width of G is at most k is NP-complete. This solves a problem that has been open since the introduction of clique-width in the early 1990s.
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