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Linear Time Solvable Optimization Problems on Graphs of Bounded Clique Width

(Extended Abstract)
  • B. Courcelle
  • J. A. Makowsky
  • U. Rotics
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1517)

Abstract

Graphs of clique-width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k-expressions based on graph operations which use k vertex labels. In this paper we show that the (q,q-4) graphs are of clique width at most q and P4-tidy graphs are of clique-width at most 4. Furthermore, the k-expression (for k=4 or k=q) associated with such a graph can be found in linear time.

q,q-4) graphs were introduced by Babel and Olariu (1995) and extends the class of P4-sparse graphs. P4-sparse graphs were introduced by Hoàng (1985) and are widely studied because of their applications in areas such as scheduling, clustering and computational semantics. Another extension of P4-sparse graphs are the P4-tidy graphs which were introduced by Rusu (1995).

Furthermore, we show that the class of LinEMSOL(τ1,L) optimization problems is solvable in O(f(|V|,|E|)) time on a class of graphs of clique-width at most k in which for every graph G an expression defining it can be constructed in O(f(|V|,|E|)) time. By the above this applies in particular to (q,q – 4) graphs, P4-tidy graphs and P4-sparse graphs with f linear.

Finally, we show that the above results cannot be extended to MSOL(τ2) decision and optimization problems on the vocabulary τ2 which allow edges to be considered as elements of the domains of the graphs in question, and by that, allow quantifying over edges in addition to quantifying over vertices.

Keywords

Internal Node Steiner Tree Prime Graph Graph Grammar Unary Predicate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • B. Courcelle
    • 1
  • J. A. Makowsky
    • 2
  • U. Rotics
    • 2
  1. 1.Laboratoire d’InformatiqueUniversité Bordeaux-ITalenceFrance
  2. 2.Department of Computer ScienceTechnion, Israel Institute of TechnologyHaifaIsrael

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