Rank-width and Tree-width of H-minor-free Graphs

TL;DR

This paper establishes bounds relating rank-width and tree-width in H-minor-free graphs, providing new insights into their structural properties and refining existing bounds for specific graph classes.

Contribution

It proves that for fixed r, the tree-width of K_r-topological minor-free graphs is linearly bounded by their rank-width, with refinements for other classes.

Findings

Tree-width is linearly bounded by rank-width in K_r-topological minor-free graphs.

Polynomial bounds are established for graphs excluding K_r as a subgraph.

Refinements are provided for graphs of bounded genus and K_r-minor free graphs.

Abstract

We prove that for any fixed r>=2, the tree-width of graphs not containing K_r as a topological minor (resp. as a subgraph) is bounded by a linear (resp. polynomial) function of their rank-width. We also present refinements of our bounds for other graph classes such as K_r-minor free graphs and graphs of bounded genus.