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.