双宽八:圈定共赢

Édouard Bonnet, Dibyayan Chakraborty, Eun Jung Kim, N. Köhler, Raul Lopes, Stéphan Thomassé
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引用次数: 22

摘要

我们引入了圈定的概念。一个图类$\mathcal C$被描述为:对于$\mathcal C$的子类$\mathcal D$的每一个遗传闭包$\mathcal D$,当且仅当$\mathcal D$是单基相关的,它保持$\mathcal D$具有有界的双宽度。对类$\mathcal C$的描述的有效强化意味着对$\mathcal C$的可处理的FO模型检查是完全可以理解的:在$\mathcal C$子类的遗传闭包$\mathcal D$上,当$\mathcal D$具有有界的双宽度时,FO模型检查是固定参数可处理的(FPT)。有序图[BGOdMSTT, STOC '22]和置换图[BKTW, JACM '22]可以有效地圈定,而亚三次图则不能。一方面,我们证明了区间图,甚至有根有向路径图的刻画。另一方面,我们证明了简单多边形的段图、有向路径图和可见性图没有被描绘出来。为了绘制区间图(已圈定)和轴平行两长段图(未圈定)之间的圈定边界,我们研究了限制段相交类的双宽度。已知(无三角形)纯轴平行单元段图具有无界双宽[BGKTW, SODA '21]。我们证明了$K_{t,t}$自由的单元段图和$H_t$自由的轴平行单元段图具有有界的双宽,其中$H_t$是高度$t$的半图或阶梯。相反,轴平行$H_4$自由的两长段图具有无界的双宽度。我们的新结果,与已知的FPT算法相结合,用于对给定O(1)$-序列的图进行FO模型检查,导致双赢的论点。例如,我们在1.5D地形的可见性图上推导了$k$-Ladder的FPT算法,在简单多边形的可见性图上推导了$k$-Independent Set的FPT算法。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Twin-width VIII: delineation and win-wins
We introduce the notion of delineation. A graph class $\mathcal C$ is said delineated if for every hereditary closure $\mathcal D$ of a subclass of $\mathcal C$, it holds that $\mathcal D$ has bounded twin-width if and only if $\mathcal D$ is monadically dependent. An effective strengthening of delineation for a class $\mathcal C$ implies that tractable FO model checking on $\mathcal C$ is perfectly understood: On hereditary closures $\mathcal D$ of subclasses of $\mathcal C$, FO model checking is fixed-parameter tractable (FPT) exactly when $\mathcal D$ has bounded twin-width. Ordered graphs [BGOdMSTT, STOC '22] and permutation graphs [BKTW, JACM '22] are effectively delineated, while subcubic graphs are not. On the one hand, we prove that interval graphs, and even, rooted directed path graphs are delineated. On the other hand, we show that segment graphs, directed path graphs, and visibility graphs of simple polygons are not delineated. In an effort to draw the delineation frontier between interval graphs (that are delineated) and axis-parallel two-lengthed segment graphs (that are not), we investigate the twin-width of restricted segment intersection classes. It was known that (triangle-free) pure axis-parallel unit segment graphs have unbounded twin-width [BGKTW, SODA '21]. We show that $K_{t,t}$-free segment graphs, and axis-parallel $H_t$-free unit segment graphs have bounded twin-width, where $H_t$ is the half-graph or ladder of height $t$. In contrast, axis-parallel $H_4$-free two-lengthed segment graphs have unbounded twin-width. Our new results, combined with the known FPT algorithm for FO model checking on graphs given with $O(1)$-sequences, lead to win-win arguments. For instance, we derive FPT algorithms for $k$-Ladder on visibility graphs of 1.5D terrains, and $k$-Independent Set on visibility graphs of simple polygons.
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