Computing the Clique-Width of Large Path Powers in Linear Time via a New Characterisation of Clique-Width

Clique-width is one of the most important graph parameters, as many NP-hard graph problems are solvable in linear time on graphs of bounded clique-width. Unfortunately the computation of clique-width is among the hardest problems. In fact we do not know of any other algorithm than brute force for the exact computation of clique-width on any nontrivial large graph class. Another difficulty about clique-width is the lack of alternative characterisations of it that might help in coping with its hardness. In this paper we present two results. The first is a new characterisation of clique-width based on rooted binary trees, completely without the use of labelled graphs. Our second result is the exact computation of the clique-width of large path powers in polynomial time, which has been an open problem since the 1990’s. The presented new characterisation is used to achieve this latter result. With our result, large k-path powers constitute the first non-trivial infinite class of graphs of unbounded clique-width whose clique-width can be computed exactly in polynomial time.