On rank-width of even-hole-free graphs

We present a class of (diamond, even hole)-free graphs with no clique cutset that has unbounded rank-width. In general, even-hole-free graphs have unbounded rank-width, because chordal graphs are even-hole-free. A. A. da Silva, A. Silva and C. Linhares-Sales (2010) showed that planar even-hole-free graphs have bounded rank-width, and N. K. Le (2016) showed that even-hole-free graphs with no star cutset have bounded rank width. A natural question is to ask, whether even-hole-free graphs with no clique cutsets have bounded rank-width. Our result gives a negative answer. Hence we cannot apply the meta-theorem by Courcelle, Makowsky and Rotics, which would provide efficient algorithms for a large number of problems, including the maximum independent set problem, whose complexity remains open for (diamond, even hole)-free graphs

A class of graphs with large rankwidth

We describe several graphs of arbitrarily large rankwidth (or equivalently of arbitrarily large cliquewidth). Korpelainen, Lozin, and Mayhill [Split permutation graphs, {\em Graphs and Combinatorics}, 30(3):633--646, 2014] proved that there exist split graphs with Dilworth number~2 of arbitrarily large rankwidth, but without explicitly constructing them. Our construction provides an explicit construction. Maffray, Penev, and Vu\v{s}kovi\'c [Coloring rings, arXiv:1907.11905, 2019] proved that graphs that they call rings on $n$ sets can be colored in polynomial time. Our construction shows that for some fixed integer $n\geq 3$, there exist rings on $n$ sets of arbitrarily large rankwidth. When $n\geq 5$ and $n$ is odd, this provides a new construction of even-hole-free graphs of arbitrarily large rankwidth

(Theta, triangle)-free and (even hole, K4K_4)-free graphs. Part 1 : Layered wheels

We present a construction called layered wheel. Layered wheels are graphs of arbitrarily large treewidth and girth. They might be an outcome for a possible theorem characterizing graphs with large treewidth in terms of their induced subgraphs (while such a characterization is well-understood in terms of minors). They also provide examples of graphs of large treewidth and large rankwidth in well-studied classes, such as (theta, triangle)-free graphs and even-hole-free graphs with no $K_4$ (where a hole is a chordless cycle of length at least four, a theta is a graph made of three internally vertex disjoint paths of length at least two linking two vertices, and $K_4$ is the complete graph on four vertices)

On rank-width of even-hole-free graphs

We present a class of (diamond, even hole)-free graphs with no clique cutset that has unbounded rank-width. In general, even-hole-free graphs have unbounded rank-width, because chordal graphs are even-hole-free. A.A. da Silva, A. Silva and C. Linhares-Sales (2010) showed that planar even-hole-free graphs have bounded rank-width, and N.K. Le (2016) showed that even-hole-free graphs with no star cutset have bounded rank-width. A natural question is to ask, whether even-hole-free graphs with no clique cutsets have bounded rank-width. Our result gives a negative answer. Hence we cannot apply Courcelle and Makowsky's meta-theorem which would provide efficient algorithms for a large number of problems, including the maximum independent set problem, whose complexity remains open for (diamond, even hole)-free graphs