Leaf Powers and Their Properties: Using the Trees View Full Text


Ontology type: schema:Chapter     


Chapter Info

DATE

2008

AUTHORS

Michael R. Fellows , Daniel Meister , Frances A. Rosamond , R. Sritharan , Jan Arne Telle

ABSTRACT

A graph G on n vertices is a k-leaf power () if it is isomorphic to a graph that can be “generated” from a tree T that has n leaves, by taking the leaves to represent vertices of G, and making two vertices adjacent if and only if they are at distance at most k in T. We address two questions in this paper: (1) As k increases, do we always have ? Answering an open question of Andreas Brandstädt and Van Bang Le [2,3,1], we show that the answer, perhaps surprisingly, is “no.” (2) How should one design algorithms to determine, for k-leaf powers, if they have some property? One way this can be done is to use the fact that k-leaf powers have bounded cliquewidth. This fact, plus the FPT cliquewidth approximation algorithm of Oum and Seymour [14], combined with the results of Courcelle, Makowsky and Rotics [7], allows us to conclude that properties expressible in a general logic formalism, can be decided in FPT time for k-leaf powers, parameterizing by k. This is wildly inefficient. We explore a different approach, under the assumption that a generating tree is given with the graph. We show that one can use the tree directly to decide the property, by means of a finite-state tree automaton. (A more general theorem has been independently obtained by Blumensath and Courcelle [5].) We place our results in a general context of “tree-definable” graph classes, of which k-leaf powers are one particular example. More... »

PAGES

402-413

References to SciGraph publications

  • 2007. On (k,ℓ)-Leaf Powers in MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE 2007
  • 2008. On k- Versus (k + 1)-Leaf Powers in COMBINATORIAL OPTIMIZATION AND APPLICATIONS
  • 1999. Parameterized Complexity in NONE
  • 2007. The Clique-Width of Tree-Power and Leaf-Power Graphs in GRAPH-THEORETIC CONCEPTS IN COMPUTER SCIENCE
  • 2000-04. Linear Time Solvable Optimization Problems on Graphs of Bounded Clique-Width in THEORY OF COMPUTING SYSTEMS
  • Book

    TITLE

    Algorithms and Computation

    ISBN

    978-3-540-92181-3
    978-3-540-92182-0

    Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/978-3-540-92182-0_37

    DOI

    http://dx.doi.org/10.1007/978-3-540-92182-0_37

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1039610614


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