Boolean-width is a recently introduced graph invariant. Similar to tree-width, it measures 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*(2 O(k2)). We relate the boolean-width of a graph to its branch-width 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 rank-width in [S. Oum. Rank-width is less than or equal to branch-width. Journal of Graph Theory 57(3):239-244, 2008]. For an n-vertex random graph, with a uniform edge distribution, we show that almost surely its boolean-width is Θ(log2 n) - setting boolean-width 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 quasi-polynomial time O*(2O(log4 n)). © 2010 Springer-Verlag.
CITATION STYLE
Adler, I., Bui-Xuan, B. M., Rabinovich, Y., Renault, G., Telle, J. A., & Vatshelle, M. (2010). On the boolean-width of a graph: Structure and applications. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 6410 LNCS, pp. 159–170). https://doi.org/10.1007/978-3-642-16926-7_16
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