Fast algorithms for vertex subset and vertex partitioning problems on graphs of low boolean-width⋆

We consider the graph parameter boolean-width, related to the number of different unions of neighborhoods across a cut of a graph. Boolean-width is similar to rankwidth, 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. It compares well to the other four well-known width parameters tree-width, branch-width, clique-width, and rank-width: for many graph classes boolean-width is bounded whereas tree-width and branch-width are unbounded; for some graph classes boolean-width has been shown to be exponentially smaller than any of the other four; for arbitrary graphs, boolean-width is never larger than branchwidth (except for extreme values of zero and one), nor tree-width plus one, nor clique-width, and has been shown to be at least smaller than the square of rank-width. Boolean-width has been shown to be a very natural parameter to consider when solving Maximum Independent Set and Minimum Dominating Set using a divide-and-conquer approach. In this paper we investigate which are the graph problems having the same behaviour, and extend them to a large class of NP-hard vertex subset and vertex partitioning problems by giving algorithms that are FPT when parameterized by either boolean-width, rank-width or clique-width, with runtime single exponential in either parameter if given the pertinent optimal decomposition.