On rank-width of even-hole-free graphs

TL;DR

This paper demonstrates that even-hole-free graphs without clique cutsets can have unbounded rank-width, showing limitations in applying certain algorithmic meta-theorems to these graph classes.

Contribution

It provides a counterexample class of (diamond, even hole)-free graphs with no clique cutset exhibiting unbounded rank-width, answering an open question in the field.

Findings

Constructed a class of graphs with unbounded rank-width

Showed that even-hole-free graphs can have unbounded rank-width

Implication that certain algorithmic approaches are not universally applicable

Abstract

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.