具有与不具有乘积结构的相交图
Intersection Graphs with and without Product Structure
AI总结：
本文研究平面相交图子类何时具有乘积结构，针对 $\alpha$-自由位似副本定义阈值 $\alpha^*(S)$，证明三角形和梯形等集合的阈值为1，而无穷多正 $n$ 边形的阈值严格介于0和1之间。
摘要：
一个图类 $\mathcal{G}$ 具有乘积结构，如果存在一个常数 $k$，使得每个 $G \in \mathcal{G}$ 都是 $H \boxtimes P$ 的子图，其中 $P$ 是一条路径，$H$ 是树宽为 $k$ 的某个图。众所周知，平面图类以及许多超平面图类都具有乘积结构。然而，我们只有很少的工具来证明乘积结构的不存在性，因此只知道少数有趣的图类例子。受乘积结构与无乘积结构之间过渡的启发，我们研究了平面中相交图的子类（例如圆盘相交图），并给出了这些图类具有乘积结构的充分必要条件。具体来说，对于集合 $S \subset \mathbb{R}^2$（例如一个圆盘）和实数 $\alpha \in [0,1]$，我们考虑 $S$ 的 $\alpha$-自由位似副本的相交图。也就是说，每个顶点 $v$ 是 $S$ 的一个位似副本，其中至少 $\alpha$ 比例的部分不被其他顶点覆盖，并且当且仅当 $u \cap v \neq \emptyset$ 时，$u$ 和 $v$ 之间有一条边。对于 $\alpha = 1$，我们得到的是接触图，它们在大多数情况下是平面的，因此具有乘积结构。对于 $\alpha = 0$，我们得到（除其他外）所有完全图，因此没有乘积结构。一般来说，存在一个阈值 $\alpha^*(S) \in [0,1]$，使得 $S$ 的 $\alpha$-自由位似副本对所有 $\alpha > \alpha^*(S)$ 具有乘积结构，而对所有 $\alpha < \alpha^*(S)$ 不具有乘积结构。我们证明，对于一大类集合 $S$，包括所有三角形和所有梯形，有 $\alpha^*(S) = 1$，即除了接触图（当 $\alpha = 1$ 时）之外，我们没有乘积结构。对于其他集合 $S$，包括无穷多个 $n$ 值对应的正 $n$ 边形，我们通过证明上界和下界来证明 $0 < \alpha^*(S) < 1$。
英文摘要：
A graph class $\mathcal{G}$ admits product structure if there exists a constant $k$ such that every $G \in \mathcal{G}$ is a subgraph of $H \boxtimes P$ for a path $P$ and some graph $H$ of treewidth $k$. Famously, the class of planar graphs, as well as many beyond-planar graph classes are known to admit product structure. However, we have only few tools to prove the absence of product structure, and hence know of only a few interesting examples of classes. Motivated by the transition between product structure and no product structure, we investigate subclasses of intersection graphs in the plane (e.g., disk intersection graphs) and present necessary and sufficient conditions for these to admit product structure. Specifically, for a set $S \subset \mathbb{R}^2$ (e.g., a disk) and a real number $α\in [0,1]$, we consider intersection graphs of $α$-free homothetic copies of $S$. That is, each vertex $v$ is a homothetic copy of $S$ of which at least an $α$-portion is not covered by other vertices, and there is an edge between $u$ and $v$ if and only if $u \cap v \neq \emptyset$. For $α= 1$ we have contact graphs, which are in most cases planar, and hence admit product structure. For $α= 0$ we have (among others) all complete graphs, and hence no product structure. In general, there is a threshold value $α^*(S) \in [0,1]$ such that $α$-free homothetic copies of $S$ admit product structure for all $α> α^*(S)$ and do not admit product structure for all $α< α^*(S)$. We show for a large family of sets $S$, including all triangles and all trapezoids, that it holds $α^*(S) = 1$, i.e., we have no product structure, except for the contact graphs (when $α= 1$). For other sets $S$, including regular $n$-gons for infinitely many values of $n$, we show that $0 < α^*(S) < 1$ by proving upper and lower bounds.
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