The relative clique-width of a graph

  • Authors:

  • Vadim Lozin;Dieter Rautenbach

  • Affiliations:

  • RUTCOR, Rutgers University, 640 Bartholomew road, Piscataway, NJ 08854-8003, USA;Institut für Mathematik, TU Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2007

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Abstract

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.