On strongly circular-perfectness

Perfect graphs constitute a well-studied graph class with a rich structure. Circular-perfect graphs as introduced by Zhu are a natural superclass of perfect graphs defined by means of a more general coloring concept and form an important class of χ-bound graphs with the smallest non-trivial χ-binding function χ ( G ) ⩽ ω ( G ) + 1 . Apart from perfect graphs, circular-perfect graphs include all convex-round graphs and outerplanar graphs. A linear description of facets of stable set polytopes of circular-perfect graphs is still unknown, though convex-round graphs and outerplanar graphs are rank-perfect. s paper, we exhibit an infinite class of non rank-perfect circular-perfect graphs. We introduce strongly circular-perfectness: a circular-perfect graph is said to be strongly circular-perfect if its complement is also circular-perfect. This subclass of circular-perfect graphs includes perfect graphs, odd holes and antiholes. We show that there are infinitely many non rank-perfect strongly circular-perfect graphs and fully characterize triangle-free minimal strongly circular-imperfect graphs: it turns out that these graphs are very close to odd holes.