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- W2926202237 abstract "A metric graph is a pair $(G,d)$, where $G$ is a graph and $d:E(G) tomathbb{R}_{geq0}$ 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 $vwin E(G)$. 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 Mobius ladder with one `horizontal edge' removed." @default.
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- W2926202237 date "2019-04-05" @default.
- W2926202237 modified "2023-09-27" @default.
- W2926202237 title "Unavoidable minors for graphs with large $ell_p$-dimension" @default.
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