Graphs of bounded depth-$2$ rank-brittleness

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.