几何交图的收缩二维性
Contraction Bidimensionality of Geometric Intersection Graphs
摘要：
给定一个图$G$，我们定义${\bf bcg}(G)$为$G$可以收缩到均匀三角网格$Γ_{k}$的最小$k$。图类${\cal G}$具有SQG ${\bf C}$性质，如果每个图$G\in{\cal G}$对于某些$1\leq c<2$具有树宽$\mathcal{O}({\bf bcg}(G)^{c})$。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法结果的适用范围，在二维理论的框架中，与快速参数化算法，核化和近似方案相关。这些结果适用于一个广泛的家庭的问题，即收缩二维的问题。我们的主要组合结果揭示了满足SQG ${\bf C}$性质的广泛的图类家族。这个族特别包括有界度弦图。这大大扩展了二维收缩问题的二维理论的适用性。
英文摘要：
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.
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