Algorithms for vertex-partitioning problems on graphs with fixed clique-width

  • Authors:

  • Michael U. Gerber;Daniel Kobler

  • Affiliations:

  • Department of Mathematics, Swiss Federal Institute of Technology, 1015 Lausanne, Switzerland;Fields Institute for Research in Mathematical Sciences, 222 College Street, Toronto, Ont., Canada M5T 3J1

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2003

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Abstract

Many vertex-partitioning problems can be expressed within a general framework introduced by Telle and Proskurowski. They showed that optimization problems in this framework can be solved in polynomial time on classes of graphs with bounded tree-width. In this paper, we consider a very similar framework, in relationship with more general classes of graphs: we propose a polynomial time algorithm on classes of graphs with bounded clique-width for all the optimization problems in our framework. These classes of graphs are more general than the classes of graphs with bounded tree-width in the sense that classes of graphs with bounded tree-width have also bounded clique-width (but not necessarily the inverse).Our framework includes problems such as independent (dominating) set, p-dominating set, induced bounded degree subgraph, induced p-regular subgraph, perfect matching cut, graph k-coloring and graph list-k-coloring with cardinality constraints (fixed k). This paper thus provides a second (distinct) framework within which the optimization problems can be solved in polynomial time on classes of graphs with bounded clique-width, after a first framework (called MS1) due to the work of Courcelle, Makowsky and Rotics (for which they obtained a linear time algorithm).