Minimal determining sets for certain $W$-graph ideals

We consider Kazhdan-Lusztig cells of the symmetric group $S_n$ containing the longest element of a standard parabolic subgroup of $S_n$. Extending some of the ideas in [Beiträge zur Algebra und Geometrie, 59 (2018) no.~3 523--547] and [Journal of Algebra and Its Applications, 20 (2021) no.~10 2150181], we determine the rim of some additional families of cells and also of certain induced unions of cells. These rims provide minimal determining sets for certain $W$-graph ideals introduced in [Journal of Algebra, 361 (2012) 188--212].