Minors and dimension

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• 作者：Bartosz Walczak¹ ; ^{walczak@tcs.uj.edu.pl}
• 关键词：Dimension ; Posets ; Cover graphs ; Structural graph theory
• 刊名：Journal of Combinatorial Theory, Series B
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：122
• 期：Complete
• 页码：668-689
• 全文大小：546 K
• 卷排序：122

文摘

It has been known for 30 years that posets with bounded height and with cover graphs of bounded maximum degree have bounded dimension. Recently, Streib and Trotter proved that dimension is bounded for posets with bounded height and planar cover graphs, and Joret et al. proved that dimension is bounded for posets with bounded height and with cover graphs of bounded tree-width. In this paper, it is proved that posets of bounded height whose cover graphs exclude a fixed topological minor have bounded dimension. This generalizes all the aforementioned results and verifies a conjecture of Joret et al. The proof relies on the Robertson–Seymour and Grohe–Marx graph structure theorems.