Triangle-free strongly circular-perfect graphs

Zhu [15] introduced circular-perfect graphs as a superclass of the well-known perfect graphs and as an important χ-bound class of graphs with the smallest non-trivial χ-binding function χ(G) ≤ ω(G)+1. Perfect graphs have been recently characterized as those graphs without odd holes and odd antiholes as induced subgraphs [4]; in particular, perfect graphs are closed under complementation [7]. In contrary, circular-perfect graphs are not closed under complementation and the list of forbidden subgraphs is unknown. We study strongly circular-perfect graphs: a circular-perfect graph is strongly circularperfect if its complement is circular-perfect as well. This subclass entails perfect graphs, odd holes, and odd antiholes. As main result, we fully characterize the triangle-free strongly circular-perfect graphs, and prove that, for this graph class, both the stable set problem and the recognition problem can be solved in polynomial time. Moreover, we address the characterization of strongly circular-perfect graphs by means of forbidden subgraphs. Results from [9] suggest that formulating a corresponding conjecture for circular-perfect graphs is difficult; it is even unknown which triangle-free graphs are minimal circular-imperfect. We present the complete list of all triangle-free minimal not strongly circular-perfect graphs. Preprint submitted to Discrete Mathematics 19 December 2007