Excluding subdivisions of bounded degree graphs☆,☆☆

https://doi.org/10.1016/j.jctb.2018.05.001Get rights and content
Under an Elsevier user license
open archive

Abstract

Let H be a fixed graph. What can be said about graphs G that have no subgraph isomorphic to a subdivision of H? Grohe and Marx proved that such graphs G satisfy a certain structure theorem that is not satisfied by graphs that contain a subdivision of a (larger) graph H1. Dvořák found a clever strengthening—his structure is not satisfied by graphs that contain a subdivision of a graph H2, where H2 has “similar embedding properties” as H. Building upon Dvořák's theorem, we prove that said graphs G satisfy a similar structure theorem. Our structure is not satisfied by graphs that contain a subdivision of a graph H3 that has similar embedding properties as H and has the same maximum degree as H. This will be important in a forthcoming application to well-quasi-ordering.

Keywords

Graph subdivision
Structure theorem
Excluded subdivision theorem

Cited by (0)

☆

Partially supported by NSF under Grant No. DMS-1202640.

☆☆

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1664593.

1

Current address: Department of Mathematics, Princeton University, Princeton, NJ 08544-1000, USA.

2

Partially supported by NSF under Grant No. DMS-1664593.

© 2018 Elsevier Inc.