Boolean-width of graphs

Abstract

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 ( n 2 k 2 2 k ) and Minimum Weight Dominating Set in time O ( n 2 + n k 2 3 k ) . With an additional n 2 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 G F ( 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 ( n 2 ) vertices has boolean-width Θ ( log n ) and rank-width, clique-width, tree-width, and branch-width Θ ( n ) .