On the Boolean-Width of a Graph: Structure and Applications

  • Isolde Adler
  • Binh-Minh Bui-Xuan
  • Yuri Rabinovich
  • Gabriel Renault
  • Jan Arne Telle
  • Martin Vatshelle
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6410)

Abstract

Boolean-width is a recently introduced graph invariant. Similar to tree-width, it measures the structural complexity of graphs. Given any graph G and a decomposition of G of boolean-width k, we give algorithms solving a large class of vertex subset and vertex partitioning problems in time \(O^*(2^{O(k^2)})\). We relate the boolean-width of a graph to its branch-width and to the boolean-width of its incidence graph. For this we use a constructive proof method that also allows much simpler proofs of similar results on rank-width in [S. Oum. Rank-width is less than or equal to branch-width. Journal of Graph Theory 57(3):239–244, 2008]. For an n-vertex random graph, with a uniform edge distribution, we show that almost surely its boolean-width is Θ(log2 n) – setting boolean-width apart from other graph invariants – and it is easy to find a decomposition witnessing this. Combining our results gives algorithms that on input a random graph on n vertices will solve a large class of vertex subset and vertex partitioning problems in quasi-polynomial time \(O^*(2^{O(\log ^4 n)})\).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Isolde Adler
    • 1
  • Binh-Minh Bui-Xuan
    • 1
  • Yuri Rabinovich
    • 2
  • Gabriel Renault
    • 1
  • Jan Arne Telle
    • 1
  • Martin Vatshelle
    • 1
  1. 1.Department of InformaticsUniversity of BergenNorway
  2. 2.Department of Computer ScienceHaifa UniversityIsrael

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