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. It is
widely believed that determining the clique-width of a graph is
NP-hard; in spite of considerable efforts, no NP-hardness proof has
been found so far. We give the first hardness proof. 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.

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. It is
widely believed that determining the clique-width of a graph is
NP-hard; in spite of considerable efforts, no NP-hardness proof has
been found so far. We give the first hardness proof. We show that
the clique-width of a graph cannot be absolutely approximated in
polynomial time unless P=NP, this solves a problem that has been open
since the introduction of clique-width in the early 1990s.