The relative clique-width of a graph

The tree-width of graphs is a well-studied notion the importance of which is partly due to the fact that many hard algorithmic problems can be solved efficiently when restricted to graphs of bounded tree-width. The same is true for the clique-width which is a relatively young notion generalizing tree-width in the sense that graphs of bounded tree-width have bounded clique-width.Whereas tree-decompositions that are used to define tree-width are a very intuitive and easily visualizable way to represent the global structure of a graph, the clique-width is much harder to grasp intuitively. To better understand the nature of the clique-width, we introduce the notion of relative clique-width and study two algorithmical problems related to it. In conjunction, these problems would allow to determine the clique-width.For one of the problems, which is to determine the relative clique-width, we propose a polynomial-time factor 2 approximation algorithm and also show an exact solution in a natural special case. The study of the other problem has brought us to an alternative and transparent proof of the known fact that graphs of bounded tree-width have bounded clique-width.