We introduce the graph parameter boolean-width, related to the number of different unions of neighborhoods–Boolean sums 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. For an n-vertex graph given with a decomposition tree of boolean-width k, we solve Maximum Weight Independent Set in time O(n²k2^2k) and Minimum Weight Dominating Set in time O(n²+nk2^3k). With an additional n² factor in the runtime, we can also count all independent sets and dominating sets of each cardinality. Boolean-width is bounded on the same classes of graphs as clique-width. boolean-width is similar to rank-width, which is related to the number of GF(2)-sums of neighborhoods instead of the Boolean sums used for boolean-width. We show for any graph that its boolean-width is at most its clique-width and at most quadratic in its rank-width. We exhibit a class of graphs, the Hsu-grids, having the property that a Hsu-grid on Θ(n²) vertices has boolean-width Θ(logn) and rank-width, clique-width, tree-width, and branch-width Θ(n).