The excluded minors for isometric realizability in the plane

Let $G$ be a graph and $p \in [1, \infty]$. The parameter $fp(G)$ is the least integer $k$ such that for all $m$ and all vectors $(rv){v \in V(G)} \subseteq \mathbb{R}^m$, there exist vectors $(qv){v \in V(G)} \subseteq \mathbb{R}^k$ satisfying $$|rv-rw|p=|qv-qw|p, \ \text{ for all }\ vw\in E(G).$$ It is easy to check that $fp(G)$ 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 $fp(G) \leq k$. 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 $K4$ along an edge and then deleting that edge. We also show that the same two graphs are the complete set of excluded minors for $f1(G) \leq 2$. 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.