平面上等距可实现的排除子式
The excluded minors for isometric realizability in the plane
摘要：
设$G$是一个图，而$p \in [1, \infty]$ .参数$f_p(G)$是最小整数$k$，使得对于所有$m$和所有向量$(r_v)_{v \in V(G)} \subseteq \mathbb{R}^m$，存在满足$$\|r_v-r_w\|_p=\|q_v-q_w\|_p, \ \text{ for all }\ vw\in E(G).$$的向量$(q_v)_{v \in V(G)} \subseteq \mathbb{R}^k$。很容易检查$f_p(G)$总是有限的，并且它是次单调的。根据Robertson和Seymour的图子式定理，对于性质$f_p(G) \leq k$，存在有限个排除子式.本文确定了$f_\infty(G) \leq 2$的完全排除子式集。排除的两个子图是$5$顶点上的轮和通过沿边粘合$K_4$的两个副本然后删除该边而获得的图。我们还表明，相同的两个图是完整的一组排除未成年人的$f_1(G) \leq 2$。此外，我们给出了一个家庭的例子表明，$f_\infty$是无界的平面图类和$f_\infty$是没有界的树宽度的函数。
英文摘要：
Let $G$ be a graph and $p \in [1, \infty]$. The parameter $f_p(G)$ is the least integer $k$ such that for all $m$ and all vectors $(r_v)_{v \in V(G)} \subseteq \mathbb{R}^m$, there exist vectors $(q_v)_{v \in V(G)} \subseteq \mathbb{R}^k$ satisfying $$\|r_v-r_w\|_p=\|q_v-q_w\|_p, \ \text{ for all }\ vw\in E(G).$$ It is easy to check that $f_p(G)$ is always finite and that it is minor monotone. By the graph minor theorem of Robertson and Seymour, there are a finite number of excluded minors for the property $f_p(G) \leq k$. In this paper, we determine the complete set of excluded minors for $f_\infty(G) \leq 2$. The two excluded minors are the wheel on $5$ vertices and the graph obtained by gluing two copies of $K_4$ along an edge and then deleting that edge. We also show that the same two graphs are the complete set of excluded minors for $f_1(G) \leq 2$. In addition, we give a family of examples that show that $f_\infty$ is unbounded on the class of planar graphs and $f_\infty$ is not bounded as a function of tree-width.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
