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2 results

  • 1 - Clique-width for hereditary graph classes

  • Edited by Allan Lo, University of Birmingham, Richard Mycroft, University of Birmingham, Guillem Perarnau, Universitat Politècnica de Catalunya, Barcelona, Andrew Treglown, University of Birmingham
  • Book:
    Surveys in Combinatorics 2019
    Published online:
    17 June 2019
    Print publication:
    27 June 2019, pp 1-56
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  • Summary

    Clique-width is a well-studied graph parameter owing to its use in understanding algorithmic tractability: if the clique-width of a graph class G is bounded by a constant, a wide range of problems that are NP-complete in general can be shown to be polynomial-time solvable on G. For this reason, the boundedness or unboundedness of clique-width has been investigated and determined for many graph classes. We survey these results for hereditary graph classes, which are the graph classes closed under taking induced subgraphs. We then discuss the algorithmic consequences of these results, in particular for the COLOURING and GRAPH ISOMORPHISM problems. We also explain a possible strong connection between results on boundedness of clique-width and on well-quasi-orderability by the induced subgraph relation for hereditary graph classes.

  • Inductive computations on graphs defined by clique-width expressions

  • Frédérique Carrère
  • Journal:
    RAIRO - Theoretical Informatics and Applications / Volume 43 / Issue 3 / July 2009
    Published online by Cambridge University Press:
    04 April 2009, pp. 625-651
    Print publication:
    July 2009

  • Labelling problems for graphs consist in building distributeddata structures, making it possible to check a given graph property or to compute a given function, the arguments of which are vertices. For an inductively computable function D, if G is a graph with n vertices and of clique-width at mostk, where k is fixed, we can associate with each vertexx of G a piece of information (bit sequence) lab(x) of lengthO(log2(n)) such that we can compute D in constanttime, using only the labels of its arguments. The preprocessing can be done in time O(h.n)where h is the height of the syntactic tree of G.We perform an inductive computation, without usingthe tools of monadic second order logic.This enables us to give an explicit labelling scheme and toavoid constants of exponential size.