Minors and Dimension

Abstract

Streib and Trotter proved in 2012 that posets with bounded height and with planar cover graphs have bounded dimension. Recently, Joret et al. proved that the dimension is bounded for posets with bounded height whose cover graphs have 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 both the aforementioned results and verifies a conjecture of Joret et al. The proof relies on the Robertson-Seymour and Grohe-Marx structural decomposition theorems.

Planar Posets Have Dimension at Most Linear in Their Height

“…Clearly, z is smaller or equal (in P ) to all the other elements of the zig-zag path Z x 0 (y) that we traversed. We have α(z) α(y) = i − 1, by (10). We also have α(z) α(y) since z y in P and thus D(z) ⊆ D(y).…”

“…x k is a directed cycle. Then α(x 1 ) · · · α(x k ) α(x 1 ) and β(x 1 ) · · · β(x k ) β(x 1 ) by (10). Thus all these inequalities hold with equality.…”

“…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]).…”

Nowhere Dense Graph Classes and Dimension

“…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];…”