Computing the Clique-width of large path powers in linear time via a new characterisation of clique-width

  • Authors:

  • Pinar Heggernes;Daniel Meister;Udi Rotics

  • Affiliations:

  • Department of Informatics, University of Bergen, Norway;Theoretical Computer Science, University of Trier, Germany;Netanya Academic College, Netanya, Israel

  • Venue:
  • CSR'11 Proceedings of the 6th international conference on Computer science: theory and applications
  • Year:
  • 2011

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Abstract

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 large graph class of unbounded clique-width. 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 for a decade. 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.