A class of graphs with large rankwidth

TL;DR

This paper explicitly constructs graphs with arbitrarily large rankwidth, including split graphs and rings, providing new insights into graph complexity and properties related to rankwidth and cliquewidth.

Contribution

It offers explicit constructions of graphs with large rankwidth, including split graphs and rings, advancing understanding of graph parameters and their extremal properties.

Findings

Constructed split graphs with large rankwidth explicitly.

Showed rings on n sets have arbitrarily large rankwidth for fixed n.

Provided new examples of even-hole-free graphs with large rankwidth.

Abstract

We describe several graphs with arbitrarily large rankwidth (or equivalently with arbitrarily large cliquewidth). Korpelainen, Lozin, and Mayhill [Split permutation graphs, Graphs and Combinatorics, 30(3):633-646, 2014] proved that there exist split graphs with Dilworth number 2 with arbitrarily large rankwidth, but without explicitly constructing them. We provide an explicit construction. Maffray, Penev, and Vu\v{s}kovi\'c [Coloring rings, Journal of Graph Theory 96(4):642-683, 2021] proved that graphs that they call rings on $n$ sets can be colored in polynomial time. We show that for every fixed integer $n\geq 3$, there exist rings on $n$ sets with arbitrarily large rankwidth. When $n\geq 5$ and $n$ is odd, this provides a new construction of even-hole-free graphs with arbitrarily large rankwidth.