← 研踪 PaperTrailarXiv:2611.09627 · v1
What remains of the planar induced minor conjectures?
首次提交 2026-11-06 23:46
版本更新 2026-11-06 23:46
雷达观测 2026-11-09 13:22
摘要
Dallard, Milanič, and Štorgel, and independently Gartland and Lokshtanov, conjectured that for every planar graph $H$, Maximum Weight Independent Set and List $3$-Coloring are polynomial-time solvable in $H$-induced-minor-free graphs. As a structural explanation, Gartland and Lokshtanov further conjectured that such graphs admit balanced separators dominated by a constant number of vertices. Bonnet and Chang recently refuted the independent-set part of the algorithmic conjecture, by showing that Maximum Independent Set is $\mathsf{NP}$-hard in graphs excluding the $5\times5$ grid as an induced minor. We revisit both conjectures. We show that the graphs used by Bonnet and Chang exclude much simpler planar graphs, including several $16$-vertex trees, and we introduce a second, independent family of hard instances. We also prove that $3$-Coloring is $\mathsf{NP}$-hard in both settings, thus refuting the algorithmic conjecture in full, even in its quasipolynomial version, unless $\mathsf{NP}\subseteq\mathsf{QP}$. On the structural side, we identify planar graphs $H$ for which the structural conjecture holds, while the algorithmic one fails. Thus, constantly dominated balanced separators cannot by themselves explain the tractability of Maximum Weight Independent Set and List $3$-Coloring.
作者备注：26 pages, 7 figures
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v1 · 2026-11-06 23:46 · 最新观测版本
What remains of the planar induced minor conjectures?
Dallard, Milanič, and Štorgel, and independently Gartland and Lokshtanov, conjectured that for every planar graph $H$, Maximum Weight Independent Set and List $3$-Coloring are polynomial-time solvable in $H$-induced-minor-free graphs. As a structural explanation, Gartland and Lokshtanov further conjectured that such graphs admit balanced separators dominated by a constant number of vertices. Bonnet and Chang recently refuted the independent-set part of the algorithmic conjecture, by showing that Maximum Independent Set is $\mathsf{NP}$-hard in graphs excluding the $5\times5$ grid as an induced minor. We revisit both conjectures. We show that the graphs used by Bonnet and Chang exclude much simpler planar graphs, including several $16$-vertex trees, and we introduce a second, independent family of hard instances. We also prove that $3$-Coloring is $\mathsf{NP}$-hard in both settings, thus refuting the algorithmic conjecture in full, even in its quasipolynomial version, unless $\mathsf{NP}\subseteq\mathsf{QP}$. On the structural side, we identify planar graphs $H$ for which the structural conjecture holds, while the algorithmic one fails. Thus, constantly dominated balanced separators cannot by themselves explain the tractability of Maximum Weight Independent Set and List $3$-Coloring.
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