Tree Pivot-Minors and Linear Rank-Width

Tree-width and its linear variant path-width play a central role for the graph minor relation. In particular, Robertson Seymour (1983) proved that every tree~$T$, class of graphs do not contain $T$ as has bounded path-width. For pivot-minor relation, rank-width take over from tree-width As such, it is natural to examine if rank-width. We first prove this statement false whenever tree caterpillar. conjecture true are also able give partial confirmation by proving: (1) $T$, $T$-pivot-minor-free distance-hereditary only caterpillar; (2) caterpillar on at most four vertices, To our second result, we need consider $T=P_4$ $T=K_{1,3}$, but follow general strategy: show contained in some $(H_1,H_2)$-free graphs, which then have $(K_3,S_{1,2,2})$-free rank-width, strengthens known result