Shrub-depth and rank-depth are dense analogues of the tree-depth of a graph. It is well known that a graph has large tree-depth if and only if it has a long path as a subgraph. We prove an analogous statement for shrub-depth and rank-depth, which was conjectured by Hliněný et al. (2016) [11]. Namely, we prove that a graph has large rank-depth if and only if it has a vertex-minor isomorphic to a long path. This implies that for every integer t, the class of graphs with no vertex-minor isomorphic to the path on t vertices has bounded shrub-depth.

中文翻译：

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有界灌木深度和等级深度的障碍

灌木深度和等级深度是图形树深度的密集类似物。众所周知，当且仅当图作为子图具有较长的路径时，图的树深度才大。我们证明了灌木深度和等级深度的类似说法，这是由赫林涅尼等人推测的。（2016）[11]。即，我们证明了当且仅当图具有对长路径的次顶点同构时，图才具有较大的秩深度。这意味着对于每个整数t，图的类别在t顶点上都没有与顶点同构的顶点次同构，而其灌木深度是有界的。