Boolean-width of graphs
Résumé
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 $O(n(n + 2^{3k} k ))$. We show for any graph that its boolean-width is never more than the square of its rank-width. We also exhibit a class of graphs, the Hsu-grids, having the property that a Hsu-grid on $\Theta(n^2)$ vertices has boolean-width $\Theta(\log n)$ and tree-width, branch-width, clique-width and rank-width $\Theta(n)$. Moreover, any optimal rank-decomposition of such a graph will have boolean-width $\Theta(n)$, {\sl i.e.} exponential in the optimal boolean-width.