A metric graph is a pair $(G,d)$, where $G$ is a graph and $d:E(G) \to\mathbb{ R}_{ \geq0}$ is a distance function. Let $p \in [1,\infty]$ be fixed. An isometric embedding of the metric graph $(G,d)$ in $\ell_p^k = (\mathbb{ R}^k, d_p)$ is a map $\phi : V(G) \to \mathbb{ R}^k$ such that $d_p(\phi(v), \phi(w)) = d(vw)$ for all edges $vw\in E(G)$. The $\ell_p$-dimension of $G$ is the least integer $k$ such that there exists an isometric embedding of $(G,d)$ in $\ell_p^k$ for all distance functions $d$ such that $(G,d)$ has an isometric embedding in $\ell_p^K$ for some $K$. It is easy to show that $\ell_p$-dimension is a minor-monotone property. In this paper, we characterize the minor-closed graph classes $\mathcal{ C}$ with bounded $\ell_p$-dimension, for $p \in \{ 2,\infty\}$. For $p=2$, we give a simple proof that $\mathcal{ C}$ has bounded $\ell_2$-dimension if and only if $\mathcal{ C}$ has bounded treewidth. In this sense, the $\ell_2$-dimension of a graph is `tied' to its treewidth. For $p=\infty$, the situation is completely different. Our main result states that a minor-closed class $\mathcal{ C}$ has bounded $\ell_\infty$-dimension if and only if $\mathcal{ C}$ excludes a graph obtained by joining copies of $K_4$ using the $2$-sum operation, or excludes a M\"obius ladder with one `horizontal edge' removed.
Categories: math.CO cs.DM math.MG