Boolean-Width of Graphs

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 that for any graph the square root of its booleanwidth is never more than its rank-width. We also exhibit a class of graphs, the Hsu-grids, having the property that a Hsu-grid on Θ(n2 ) vertices has boolean-width Θ(log n) and tree-width, branch-width, clique-width and rank-width Θ(n). Moreover, any optimal rankdecomposition of such a graph will have bo...

• Type: Conference
• Year: 2009
• Where: IWPEC
• Authors: Binh-Minh Bui-Xuan, Jan Arne Telle, Martin Vatshelle
• Keywords: Applied Computing, Dominating Set, Graph Parameter, Graph Problems, IWPEC 2009