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We study the recently introduced boolean-width of graphs. Our structuralresults are as follows. Firstly, we show that almost surely the boolean-widthof a random graph on $n$ vertices is $O(\log^2 n)$, and it is easy to find thecorresponding decomposition tree.  Secondly, for any constant $d$ a graph of maximum degree $d$ hasboolean-width linear in treewidth. This implies that almost surely theboolean-width of a (sparse) random $d$-regular graph on $n$ vertices is linearin $n$.  Thirdly, we show that the boolean-cut value is well approximated by VCdimension of corresponding set system. Since VC dimension is widely studied, wehope that this structural result will prove helpful in better understanding ofboolean-width.  Combining our first structural result with algorithms from Bui-Xuan et al\cite{BTV09,BTV09II} we get for random graphs quasi-polynomial $O^*(2^{O(\log^4 n)})$ time algorithms for a large class of vertex subset and vertexpartitioning problems.

On the boolean-width of a graph: structure and applications