The excluded minors for isometric realizability in the plane

Let $G$ be a graph and $p\in [1,\mathrm{\infty }]$. 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$,thereexistvectors$(q_v){v \in V(G)} \subseteq $\mathbb{R}$^k$satisfying$$|r_v-r_w|_p=|q_v-q_w|_p, \ $\text{ for all }$
vw\in E(G).$$Itiseasytocheckthat$f_p(G)$ 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 $f_p(G)\le k$. In this paper, we determine the complete set of excluded minors for $f_{\mathrm{\infty }}(G)\le 2$. 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 $f_1(G)\le 2$. In addition, we give a family of examples that show that $f_{\mathrm{\infty }}$ is unbounded on the class of planar graphs and $f_{\mathrm{\infty }}$ is not bounded as a function of tree-width.