Grid induced minor theorem for graphs of small degree
A graph H is an induced minor of a graph G if H can be obtained from G by vertex deletions and edge contractions. We show that there is a function f(k,d)=O(k10+2d5) so that if a graph has treewidth at least f(k,d) and maximum degree at most d, then it contains a k×k-grid as an induced minor. This pr...
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| Published in: | Journal of combinatorial theory. Series B Vol. 160; pp. 206 - 214 |
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| Language: | English |
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01.05.2023
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| Abstract | A graph H is an induced minor of a graph G if H can be obtained from G by vertex deletions and edge contractions. We show that there is a function f(k,d)=O(k10+2d5) so that if a graph has treewidth at least f(k,d) and maximum degree at most d, then it contains a k×k-grid as an induced minor. This proves the conjecture of Aboulker, Adler, Kim, Sintiari, and Trotignon (2021) [1] that any graph with large treewidth and bounded maximum degree contains a large wall or the line graph of a large wall as an induced subgraph. It also implies that for any fixed planar graph H, there is a subexponential time algorithm for maximum weight independent set on H-induced-minor-free graphs. |
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| AbstractList | A graph H is an induced minor of a graph G if H can be obtained from G by vertex deletions and edge contractions. We show that there is a function f(k,d)=O(k10+2d5) so that if a graph has treewidth at least f(k,d) and maximum degree at most d, then it contains a k×k-grid as an induced minor. This proves the conjecture of Aboulker, Adler, Kim, Sintiari, and Trotignon (2021) [1] that any graph with large treewidth and bounded maximum degree contains a large wall or the line graph of a large wall as an induced subgraph. It also implies that for any fixed planar graph H, there is a subexponential time algorithm for maximum weight independent set on H-induced-minor-free graphs. |
| Author | Korhonen, Tuukka |
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| Cites_doi | 10.1006/jctb.2000.2026 10.1016/j.jctb.2011.02.008 10.1145/2820609 10.1016/0095-8956(86)90030-4 10.1016/j.jctb.2022.05.009 10.1137/S0097539793251219 10.1016/j.ejc.2021.103394 10.1016/0890-5401(90)90043-H 10.1016/0166-218X(89)90031-0 10.1016/j.jctb.2020.09.010 |
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| SubjectTerms | Grid Induced minor Treewidth |
| Title | Grid induced minor theorem for graphs of small degree |
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