Unavoidable Minors for Graphs with Large $\ell _p$-Dimension

Abstract

A metric graph is a pair (G, d), where G is a graph and $d:E(G) \rightarrow \mathbb {R}_{\ge 0}$ 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 $\phi :V(G) \rightarrow \mathbb {R}^k$ 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 $\mathscr {C}$ with bounded $\ell _p$-dimension, for $p \in \{2,\infty \}$. For $p=2$, we give a simple proof that $\mathscr {C}$ has bounded $\ell _2$-dimension if and only if $\mathscr {C}$ 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 $\mathscr {C}$ has bounded $\ell _\infty$-dimension if and only if $\mathscr {C}$ excludes a graph obtained by joining copies of $K_4$ using the 2-sum operation, or excludes a Möbius ladder with one ‘horizontal edge’ removed.