  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.
