We show that Max Independent Set (MIS) remains NP-hard in graphs excluding the $5 \times 5$ grid as an induced minor. This negatively answers a question of Dallard--Milanič--Štorgel that has attracted sustained attention, and refutes a conjecture by Gartland and Lokshtanov. Unless $N$-variable 3-SAT can be solved in time $2^{\mathcal O(N^{1-\varepsilon})}$ for some $\varepsilon > 0$, MIS requires $2^{Ω(n^{1/2-o(1)})}$ time on $n$-vertex graphs in this class. This complements a $2^{\widetilde{O}(n^{2/3})}$-time algorithm on any $H$-induced-minor-free class by Korhonen and Lokshtanov. Exploiting a connection between low twin-width or reduced parameters on one hand and planar-induced-minor-freeness on the other, we further show that 3-Coloring, Feedback Vertex Set and Dominating Set are NP-complete on weakly sparse graphs of reduced component max-leaf (hence twin-width) at most 3, even when a witnessing contraction sequence is given. Classes of bounded reduced component max-leaf (by any constant) and of twin-width at most 3 are known to exclude a fixed planar graph as an induced minor. The 3-Coloring and Feedback Vertex Set results refute more cases of the Gartland--Lokshtanov conjecture, even when the inputs are weakly sparse. The complexity of MIS in weakly sparse graphs excluding a planar induced minor remains open.
A hereditary graph class has strongly edge-sublinear separators if, for some constant $0 \leqslant b < 1$, every graph in the class with $m$ edges admits a balanced separator of size $\mathcal O(m^b)$. We show that Max Weight Independent Set (MWIS) admits a deterministic quasipolynomial-time approximation scheme (QPTAS) on every such class. In particular, this applies to graphs excluding any fixed graph as an induced minor, by a separator theorem of Korhonen and Lokshtanov [SODA 2024]. Our theorem generalizes the QPTASes for axis-parallel rectangles [Adamaszek and Wiese, FOCS 2013] and polygons [Adamaszek, Har-Peled, and Wiese, JACM 2019], or equivalently for packing connected subgraphs of a planar graph [Pilipczuk, van Leeuwen, and Wiese, Algorithmica 2020]. It also generalizes the QPTAS for MWIS on hereditary classes with strongly sublinear separators measured in vertices [Dvořák, EJC 2018; Dvořák and Lahiri, ESA 2021], as well as the QPTASes for graphs excluding specific induced minors [Chudnovsky et al., arXiv 2025; Bonnet et al., SWAT 2026; Czyżewska et al., arXiv 2026].