论文标题
几何相交图的收缩二维
Contraction Bidimensionality of Geometric Intersection Graphs
论文作者
论文摘要
给定图表$ g $，我们将$ {\ bf bcg}（g）$定义为可以将$ g $签给均匀三角形的网格$γ_{k} $的最低$ k $。 Graph类$ {\ cal G} $具有sqg $ {\ bf c} $属性，如果每个图$ g \ in {\ cal g} $ in {\ cal g} $都有treewidth $ \ Mathcal {o}（{\ bf bcg bcg}（g）（g）（g）^{c}）$，对于某些$ 1 \ 1 \ leq c c <2 $。 SQG $ {\ bf c} $属性对于算法设计很重要，因为它定义了一系列元算力结果的适用性范围，在辅助性理论的框架中，与快速参数化的算法，kernelization和近似方案有关。这些结果适用于广泛的问题，即收缩二维问题。我们的主要组合结果揭示了满足SQG $ {\ bf C} $属性的广泛的图形类。这个家族特别是有界度的字符串图。这大大扩展了双二维理论在收缩二维问题中的适用性。
Given a graph $G$, we define ${\bf bcg}(G)$ as the minimum $k$ for which $G$ can be contracted to the uniformly triangulated grid $Γ_{k}$. A graph class ${\cal G}$ has the SQG${\bf C}$ property if every graph $G\in{\cal G}$ has treewidth $\mathcal{O}({\bf bcg}(G)^{c})$ for some $1\leq c<2$. The SQG${\bf C}$ property is important for algorithm design as it defines the applicability horizon of a series of meta-algorithmic results, in the framework of bidimensionality theory, related to fast parameterized algorithms, kernelization, and approximation schemes. These results apply to a wide family of problems, namely problems that are contraction-bidimensional. Our main combinatorial result reveals a wide family of graph classes that satisfy the SQG${\bf C}$ property. This family includes, in particular, bounded-degree string graphs. This considerably extends the applicability of bidimensionality theory for contraction bidimensional problems.
