← 研踪 PaperTrailarXiv:2609.11285 · v2
Max Independent Set Remains NP-hard when Excluding a Planar Induced Minor (and More)
首次提交 2026-09-10 17:17
版本更新 2026-11-03 19:28
雷达观测 2026-11-04 23:39
摘要
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.
作者备注：34 pages, 11 figures
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v2 · 2026-11-03 19:28 · 最新观测版本
Max Independent Set Remains NP-hard when Excluding a Planar Induced Minor (and More)
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.
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