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