The Excluded Minors for Isometric Realizability in the Plane
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 $\|r_v-r_w\|_p=\|q_v-q_w\|_p$ for all $vw\in E(G)....
| Main Authors: | , , , |
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| Other Authors: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
2017
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| Subjects: | |
| Online Access: | https://hdl.handle.net/10356/81454 http://hdl.handle.net/10220/43485 |
| Summary: | 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 $\|r_v-r_w\|_p=\|q_v-q_w\|_p$ 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 [J. Combin. Theory Ser. B, 92 (2004), pp. 325--357], 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 five 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. |
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