Unavoidable minors for graphs with large ℓ_p-dimension
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A metric graph is a pair (G,d), where G is a graph and d:E(G) →R_≥0 is a distance function. Let p ∈ [1,∞] be fixed. An isometric embedding of the metric graph (G,d) in ℓ_p^k = (R^k, d_p) is a map ϕ : V(G) →R^k such that d_p(ϕ(v), ϕ(w)) = d(vw) for all edges vw∈ E(G). The ℓ_p-dimension of G is the least integer k such that there exists an isometric embedding of (G,d) in ℓ_p^k for all distance functions d such that (G,d) has an isometric embedding in ℓ_p^K for some K. It is easy to show that ℓ_p-dimension is a minor-monotone property. In this paper, we characterize the minor-closed graph classes C with bounded ℓ_p-dimension, for p ∈{2,∞}. For p=2, we give a simple proof that C has bounded ℓ_2-dimension if and only if C has bounded treewidth. In this sense, the ℓ_2-dimension of a graph is `tied' to its treewidth. For p=∞, the situation is completely different. Our main result states that a minor-closed class C has bounded ℓ_∞-dimension if and only if 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.
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