具有对数布尔宽度的图类
On graph classes with logarithmic boolean-width
摘要：
布尔宽度是最近引入的图形参数。当用布尔宽度参数化时，许多问题是固定参数可处理的，例如“最小加权支配集“（MWDS）问题可以在给定宽度$k$的布尔分解的$O^*(2^{3k})$时间内求解，因此对于所有可以在多项式时间内找到宽度$O(\log n)$的布尔分解的图类，MWDS可以在多项式时间内求解。我们研究了具有布尔宽度$O(\log n)$的图类和在$O^*(2^{O(k)})$中可解的问题，结合这两个结果来设计多项式算法。我们证明了对于梯形图、循环置换图、凸图、Dilworth- $k$图、圆弧图和$k$ -退化图的补图，可以在多项式时间内找到宽度为$O(\log n)$的布尔分解.我们还证明了圆形$k$ -梯形图具有布尔宽度$O(\log n)$，并在给出圆形$k$ -梯形交模型的条件下找到了这样的分解。对于许多图类，我们还证明了它们包含布尔宽度的图$Θ(\log n)$。进一步，我们应用\cite{boolw2}的结果给出了一个新的多项式时间算法，解决了Proskurowski和Telle \cite{TP97}提出的所有顶点划分问题.这扩展了Kratochvíl，Manuel和Miller \cite{KMM95}以前的结果表明，一个大的子集的顶点划分问题是多项式可解的区间图。
英文摘要：
Boolean-width is a recently introduced graph parameter. Many problems are fixed parameter tractable when parametrized by boolean-width, for instance "Minimum Weighted Dominating Set" (MWDS) problem can be solved in $O^*(2^{3k})$ time given a boolean-decomposition of width $k$, hence for all graph classes where a boolean-decomposition of width $O(\log n)$ can be found in polynomial time, MWDS can be solved in polynomial time. We study graph classes having boolean-width $O(\log n)$ and problems solvable in $O^*(2^{O(k)})$, combining these two results to design polynomial algorithms. We show that for trapezoid graphs, circular permutation graphs, convex graphs, Dilworth-$k$ graphs, circular arc graphs and complements of $k$-degenerate graphs, boolean-decompositions of width $O(\log n)$ can be found in polynomial time. We also show that circular $k$-trapezoid graphs have boolean-width $O(\log n)$, and find such a decomposition if a circular $k$-trapezoid intersection model is given. For many of the graph classes we also prove that they contain graphs of boolean-width $Θ(\log n)$. Further we apply the results from \cite{boolw2} to give a new polynomial time algorithm solving all vertex partitioning problems introduced by Proskurowski and Telle \cite{TP97}. This extends previous results by Kratochvíl, Manuel and Miller \cite{KMM95} showing that a large subset of the vertex partitioning problems are polynomial solvable on interval graphs.
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