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Graph parameters measure the amount of structure (or lack thereof) in a graphthat makes it amenable to being decomposed in a way that facilitates dynamicprogramming. Graph decompositions and their associated parameters are importantboth in practice (as a tool for designing robust algorithms for NP-hardproblems) and in theory (relating large classes of problems to the graphs onwhich they are solvable in polynomial time).  Linear MIM-width is a variant of the graph parameter MIM-width, introduced byVatshelle. MIM-width is a parameter that is constant for many classes ofgraphs. Most graph classes which have been shown to have constant MIM-widthalso have constant linear MIM-width. However, computing the (linear) MIM-widthof graphs, or showing that it is hard, has proven to be a huge challenge. Todate, the only graph class with unbounded linear MIM-width, whose linearMIM-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-widthof its square $T^2$ always lies between $k$ and $2k$, and that these bounds aretight for all $k$.

Linear MIM-width of the Square of Trees