Abstract | ||
---|---|---|
A ladder is a 2 × k grid graph. When does a graph class
$${\cal C}$$
exclude some ladder as a minor? We show that this is the case if and only if all graphs G in
$${\cal C}$$
admit a proper vertex coloring with a bounded number of colors such that for every 2-connected subgraph H of G, there is a color that appears exactly once in H. This type of vertex coloring is a relaxation of the notion of centered coloring, where for every connected subgraph H of G, there must be a color that appears exactly once in H. The minimum number of colors in a centered coloring of G is the treedepth of G, and it is known that classes of graphs with bounded treedepth are exactly those that exclude a fixed path as a subgraph, or equivalently, as a minor. In this sense, the structure of graphs excluding a fixed ladder as a minor resembles the structure of graphs without long paths. Another similarity is as follows: It is an easy observation that every connected graph with two vertex-disjoint paths of length k has a path of length k+1. We show that every 3-connected graph which contains as a minor a union of sufficiently many vertex-disjoint copies of a 2×k grid has a 2×(k+1) grid minor. Our structural results have applications to poset dimension. We show that posets whose cover graphs exclude a fixed ladder as a minor have bounded dimension. This is a new step towards the goal of understanding which graphs are unavoidable as minors in cover graphs of posets with large dimension. |
Year | DOI | Venue |
---|---|---|
2022 | 10.1007/s00493-021-4592-8 | Combinatorica |
Keywords | DocType | Volume |
05C83, 06A07 | Journal | 42 |
Issue | ISSN | Citations |
3 | 0209-9683 | 0 |
PageRank | References | Authors |
0.34 | 11 | 5 |
Name | Order | Citations | PageRank |
---|---|---|---|
Huynh Tony | 1 | 0 | 0.68 |
Gwenaël Joret | 2 | 196 | 28.64 |
Micek Piotr | 3 | 0 | 0.34 |
Seweryn Michał T. | 4 | 0 | 0.34 |
Paul Wollan | 5 | 296 | 26.33 |