A Natural Generalization of Bounded Tree-Width and Bounded Clique-Width

We investigate a new width parameter, the fusion-width of a graph. It is a natural generalization of the tree-width, yet strong enough that not only graphs of bounded tree-width, but also graphs of bounded clique-width, trivially have bounded fusion-width. In particular, there is no exponential growth between tree-width and fusion-width, as is the case between tree-width and clique-width. The new parameter gives a good intuition about the relationship between tree-width and clique-width.Comment: To appear in the proceedings of Latin 2014. Springer LNCS 839

Line graphs of bounded clique-width

AbstractWe show that a set of graphs has bounded tree-width or bounded path-width if and only if the corresponding set of line graphs has bounded clique-width or bounded linear clique-width, respectively. This relationship implies some interesting algorithmic properties and re-proves already known results in a very simple way. It also shows that the minimization problem for NLC-width is NP-complete

Clique-width minimization is NP-hard

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 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 NPhy complete. This solves a problem that has been open since the introduction of clique-width in the early 1990s