 
Summary: On the booleanwidth of a graph: structure and applications
I. Adler1 , B.M. BuiXuan1 , Y. Rabinovich2 , G. Renault1 , J. A. Telle1 , and M. Vatshelle1
1
Department of Informatics, University of Bergen, Norway
2
Department of Computer Science, Haifa University, Israel
Abstract Booleanwidth is a recently introduced graph invariant. Similar to treewidth, it mea
sures the structural complexity of graphs. Given any graph G and a decomposition of G of boolean
width k, we give algorithms solving a large class of vertex subset and vertex partitioning problems
in time O
(2O(k2
)
). We relate the booleanwidth of a graph to its branchwidth and to the boolean
width of its incidence graph. For this we use a constructive proof method that also allows much
simpler proofs of similar results on rankwidth by Oum (JGT 2008). For a random graph on n
vertices we show that almost surely its booleanwidth is (log2
n) setting booleanwidth apart
from other graph invariants and it is easy to find a decomposition witnessing this. Combining
our results gives algorithms that on input a random graph on n vertices will solve a large class of
vertex subset and vertex partitioning problems in quasipolynomial time O
