Max Independent Set Remains NP-hard when Excluding a Planar Induced Minor (and More)

Abstract

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 $2^{\mathcal{O}(N^{1-\epsilon })}$ for some $\epsilon >0$, MIS requires $2^{\Omega (n^{1/2-o(1)})}$ time on $n$-vertex graphs in this class. This complements a $2^{\tilde{O}(n^{2/3})}$-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.

arXiv comment34 pages, 11 figures