Graph Classes with Structured Neighborhoods and Algorithmic Applications

Boolean-width is a recently introduced graph width parameter. If a boolean decomposition of width w is given, several NP-complete problems, such as Maximum Weight Independent Set, k-Coloring and Minimum Weight Dominating Set are solvable in O*(2O(w)) time [6]. In this paper we study graph classes for which we can compute a decomposition of logarithmic boolean-width in polynomial time. Since 2O(logn)=nO(1), this gives polynomial time algorithms for the above problems on these graph classes. For interval graphs we show how to construct decompositions where neighborhoods of vertex subsets are nested. We generalize this idea to neighborhoods that can be represented by a constant number of vertices. Moreover we show that these decompositions have boolean-width O(logn). Graph classes having such decompositions include circular arc graphs, circular k-trapezoid graphs, convex graphs, Dilworth k graphs, k-polygon graphs and complements of k-degenerate graphs. Combined with results in [1,5], this implies that a large class of vertex subset and vertex partitioning problems can be solved in polynomial time on these graph classes.

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