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Topological Minors of Cover Graphs and Dimension
Author(s) -
Micek Piotr,
Wiechert Veit
Publication year - 2017
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.22127
Subject(s) - mathematics , combinatorics , bounded function , discrete mathematics , treewidth , cover (algebra) , partially ordered set , split graph , graph , pathwidth , line graph , topology (electrical circuits) , mechanical engineering , mathematical analysis , engineering
We show that posets of bounded height whose cover graphs exclude a fixed graph as a topological minor have bounded dimension. This result was already proven by Walczak. However, our argument is entirely combinatorial and does not rely on structural decomposition theorems. Given a poset with large dimension but bounded height, we directly find a large clique subdivision in its cover graph. Therefore, our proof is accessible to readers not familiar with topological graph theory, and it allows us to provide explicit upper bounds on the dimension. With the introduced tools we show a second result that is supporting a conjectured generalization of the previous result. We prove that ( k + k ) ‐free posets whose cover graphs exclude a fixed graph as a topological minor contain only standard examples of size bounded in terms of k .