We characterize classes of graphs closed under taking vertex-minors and having no $P_n$ and no disjoint union of $n$ copies of the $1$-subdivision of $K_{1,n}$ for some $n$. Our characterization is described in terms of a tree of radius $2$ whose leaves are labelled by the vertices of a graph $G$, and the width is measured by the maximum possible cut-rank of a partition of $V(G)$ induced by splitting an internal node of the tree to make two components. The minimum width possible is called the depth-$2$ rank-brittleness of $G$. We prove that for all $n$, every graph with sufficiently large depth-$2$ rank-brittleness contains $P_n$ or disjoint union of $n$ copies of the $1$-subdivision of $K_{1,n}$ as a vertex-minor. This allows us to prove that for every positive integer $n$, graphs with no vertex-minor isomorphic to the disjoint union of $n$ copies of $P_5$ have bounded rank-depth and bounded linear rank-width.

中文翻译：

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有界深度 2 秩脆性图

我们描述了在取顶点次要条件下封闭的图类，并且没有 $P_n$ 并且没有 $K_{1,n}$ 的 $1$-subdivision 的 $n$ 副本的不相交联合对于某些 $n$。我们的特征是用半径为 $2$ 的树来描述的，它的叶子由图 $G$ 的顶点标记，宽度由 $V(G)$ 诱导的分区的最大可能割秩来衡量通过拆分树的内部节点以生成两个组件。可能的最小宽度称为 G$ 的深度-$2$ 等级脆性。我们证明，对于所有的 $n$，每个具有足够大的深度-$2$ rank-brittleness 的图都包含 $P_n$ 或 $K_{1,n}$ 的 $1$-细分的 $n$ 个副本的不相交并集作为小顶点。这使我们能够证明对于每个正整数 $n$，