Boolean-Width of Graphs

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+ 2k)) . We show that for any graph the square root of its boolean-width is never more than its rank-width. We also exhibit a class of graphs, the Hsugrids, having the property that a Hsu-grid on Θ(n) vertices has boolean-width Θ(log n) and tree-width, branch-width, clique-width and rank-width Θ(n) . Moreover, any optimal rank-decomposition of such a graph will have boolean-width Θ(n) , i.e. exponential in the optimal boolean-width.