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

 + 2

3

k

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 Θ(

n

2

) 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.