  Graph parameters measure the amount of structure (or lack thereof) in a graph
that makes it amenable to being decomposed in a way that facilitates dynamic
programming. Graph decompositions and their associated parameters are important
both in practice (as a tool for designing robust algorithms for NP-hard
problems) and in theory (relating large classes of problems to the graphs on
which they are solvable in polynomial time).
  Linear MIM-width is a variant of the graph parameter MIM-width, introduced by
Vatshelle. MIM-width is a parameter that is constant for many classes of
graphs. Most graph classes which have been shown to have constant MIM-width
also have constant linear MIM-width. However, computing the (linear) MIM-width
of graphs, or showing that it is hard, has proven to be a huge challenge. To
date, the only graph class with unbounded linear MIM-width, whose linear
MIM-width can be computed in polynomial time, is the trees. In this follow-up,
we show that for any tree $T$ with linear MIM-width $k$, the linear MIM-width
of its square $T^2$ always lies between $k$ and $2k$, and that these bounds are
tight for all $k$.
