A class of graphs with large rankwidth - Hoàng, Chính T.; Trotignon, Nicolas (Discrete mathematics, 2024)
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 Vušković [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≥3, there exist rings on n sets with arbitrarily large rankwidth. When n≥5 and n is odd, this provides a new construction of even-hole-free graphs with arbitrarily large rankwidth.