A ladder is a 2 × k grid graph. When does a graph class \({\cal C}\) exclude some ladder as a minor? We show that this is the case if and only if all graphs G in \({\cal C}\) admit a proper vertex coloring with a bounded number of colors such that for every 2-connected subgraph H of G, there is a color that appears exactly once in H. This type of vertex coloring is a relaxation of the notion of centered coloring, where for every connected subgraph H of G, there must be a color that appears exactly once in H. The minimum number of colors in a centered coloring of G is the treedepth of G, and it is known that classes of graphs with bounded treedepth are exactly those that exclude a fixed path as a subgraph, or equivalently, as a minor. In this sense, the structure of graphs excluding a fixed ladder as a minor resembles the structure of graphs without long paths. Another similarity is as follows: It is an easy observation that every connected graph with two vertex-disjoint paths of length k has a path of length k+1. We show that every 3-connected graph which contains as a minor a union of sufficiently many vertex-disjoint copies of a 2×k grid has a 2×(k+1) grid minor.

Our structural results have applications to poset dimension. We show that posets whose cover graphs exclude a fixed ladder as a minor have bounded dimension. This is a new step towards the goal of understanding which graphs are unavoidable as minors in cover graphs of posets with large dimension.

阶梯是一个 2 × k 的网格图。图形类\({\cal C}\)何时将某些梯子排除为次要？我们表明，这种情况下，如果仅当所有的图形和ģ在\（{\ CALÇ} \）承认的适当顶点与颜色的有限数量的着色，使得对于每个2-连通子ħ ģ，有一个在H中只出现一次的颜色。这种类型的顶点着色是居中着色的概念，其中对于每一个连通子的松弛ħ的ģ，必须有在恰好出现一次的彩色ħ。G的中心着色中的最小颜色数是G的树深度，并且已知具有有界树深度的图类正是那些将固定路径排除为子图或等价地作为次要图的图类。从这个意义上说，不包括固定阶梯作为次要的图的结构类似于没有长路径的图的结构。另一个相似之处如下：很容易观察到，每个具有两条长度为k 的顶点不相交路径的连通图都有一条长度为k +1的路径。我们表明，每个包含 2× k网格的足够多的顶点不相交副本的联合作为次要的 3-连通图具有 2×( k +1) 次要网格。

我们的结构结果适用于偏序维。我们表明，其覆盖图排除固定梯子作为次要的偏序集具有有界维度。这是朝着理解哪些图不可避免地作为大维度偏序集的覆盖图中的次要的目标迈出的新一步。