Topological Minors of Cover Graphs and Dimension

Abstract

Abstract. 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.

Nowhere Dense Graph Classes and Dimension

“…for every graph G excluding K t as a minor. Together with Theorem 2, this yields the following improvement on the previous best bound [17], which was doubly exponential in the height (for fixed t).…”

“…Regarding graphs G that exclude K t as a topological minor, it is implicitly proven in the work of Kreutzer, Pilipczuk, Rabinovich, and Siebertz [15] that these graphs satisfy wcol r (G) 2 O(r log r) when t is fixed. Combining this inequality with Theorem 2 we get a slight improvement upon the bound derived in [17], however the resulting bound remains doubly exponential: Corollary 17. Let t 1 be a fixed integer.…”

“…Our proof of Theorem 5 takes its roots in the alternative proof due to Micek and Wiechert [17] of Walczak's theorem [28], that bounded-height posets whose cover graphs exclude K t as a topological minor have bounded dimension. This proof is essentially an iterative algorithm which, if the dimension is large enough (as a function of the height), explicitly builds a subdivision of K t , one branch vertex at a time.…”

“…• have bounded treewidth, bounded genus, or more generally exclude an apex-graph as minor [9]; • exclude a fixed graph as a (topological) minor [28,17];…”

Planar Posets Have Dimension at Most Linear in Their Height

“…It soon turned out that a bound on the dimension in terms of the height holds more generally for posets with a sparse and 'well-structured' cover graph. This was shown in a series of generalizations of Theorem 7: Given a class of graphs C, every poset with a cover graph in C has dimension bounded in terms of its height if (i) C excludes an apex graph as a minor ( [2]); (ii) C excludes a fixed graph as a (topological) minor ( [10,6]); (iii) C has bounded expansion ( [4]).…”