### Boolean-Width of Graphs [chapter]

B. -M. Bui-Xuan, J. A. Telle, M. Vatshelle
2009 Lecture Notes in Computer Science
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-andconquer 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 k2 2k ) and Minimum Weight Dominating Set in time O(n 2 + nk2 3k ). With an additional n 2 factor in
more » ... 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. booleanwidth 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 rankwidth. 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).