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.

甲度量图形是一对（g ^，  d），其中G ^是一个曲线图和\（d：E（G）\ RIGHTARROW \ mathbb {R} _ {\ GE 0} \）是距离函数。令\（p \ in [1，\ infty] \）是固定的。度量图（G，  d）在\（\ ell _p ^ k =（\ mathbb {R} ^ k，d_p）\）中的等距嵌入是映射\（\ phi：V（G）\ rightarrow \ mathbb {R} ^ k \）使得对于所有边\（v（在E（G）\）中的\（d_p（\ phi（v），\ phi（w））= d（vw ）\）。的\（\ ELL _p \）-尺寸的ģ是最小整数ķ使得存在（的等距嵌入ģ，  d）在\（\ ELL _p-1K-\）对于所有的距离函数ð使得（g ^，  d）具有在等距嵌入\（\ ELL _p-1K-\）为一些 ķ。很容易证明\（\ ell _p \）-维是次要单调属性。在本文中，我们对\（p \ in \ {2，\ infty \} \）的有界\（\ ell _p \）-维来刻画次闭合图类\（\ mathscr {C } \）的特征。对于\（p = 2 \），我们给出一个简单的证明\（\ mathscr {C} \）已界\（\ ell _2 \）- dimension仅当\（\ mathscr {C} \）具有限制的树宽时才可用。从这个意义上讲，图的\（\ ell _2 \）-维“绑定”到其树宽。对于\（p = \ infty \），情况完全不同。我们的主要结果表明，当且仅当\（\ mathscr {C} \）排除通过以下方法获得的图时，次封闭类\（\ mathscr {C} \）限制了\（\ ell _ \ infty \）-维使用2和运算来连接\（K_4 \）的副本，或者排除已删除一个“水平边”的Möbius阶梯。