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

A metric graph is a pair (G, d), where G and $$d:E(G) \rightarrow \mathbb {R}_{\ge 0}$$ distance function. Let $$p \in [1,\infty ]$$ be fixed. An isometric embedding of the d) in $$\ell _p^k = (\mathbb {R}^k, d_p)$$ map $$\phi :V(G) {R}^k$$ such that $$d_p(\phi (v), \phi (w)) d(vw)$$ for all edges $$vw\in E(G)$$ . The _p$$ -dimension least integer k there exists an _p^k$$ functions d has _p^K$$ some K. It easy to show minor-monotone property. In this paper, we characterize minor-closed classes $$\mathscr {C}$$ with bounded -dimension, \{2,\infty \}$$ For $$p=2$$ , give simple proof _2$$ if only treewidth. sense, ‘tied’ its $$p=\infty $$ situation completely different. Our main result states class _\infty excludes obtained by joining copies $$K_4$$ using 2-sum operation, or Möbius ladder one ‘horizontal edge’ removed.