Boolean-Width of Graphs

  • Authors:

  • B. -M. Bui-Xuan;J. A. Telle;M. Vatshelle

  • Affiliations:

  • Department of Informatics, University of Bergen, Norway;Department of Informatics, University of Bergen, Norway;Department of Informatics, University of Bergen, Norway

  • Venue:
  • Parameterized and Exact Computation
  • Year:
  • 2009

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Abstract

We introduce the graph parameter boolean-width, related to the number of different unions of neighborhoods across a cut of a graph. For many graph problems this number is the runtime bottleneck when using a divide-and-conquer approach. Boolean-width is similar to rank-width, which is related to the number of GF(2)-sums (1+1=0) of neighborhoods instead of the Boolean-sums (1+1=1) used for boolean-width. For an n-vertex graph G given with a decomposition tree of boolean-width k we show how to solve Minimum Dominating Set, Maximum Independent Set and Minimum or Maximum Independent Dominating Set in time O(n(n + 23k k )). We show for any graph that its boolean-width is never more than the square of its rank-width. We also exhibit a class of graphs, the Hsu-grids, having the property that a Hsu-grid on 驴(n 2) vertices has boolean-width 驴(logn) and tree-width, branch-width, clique-width and rank-width 驴(n). Moreover, any optimal rank-decomposition of such a graph will have boolean-width 驴(n), i.e. exponential in the optimal boolean-width.