Obstructions for χ-diperfectness

Abstract

In 1982, Berge defined the class of χ-diperfect digraphs. A digraph D is χ-diperfect if for every minimum coloring S of D there is a path P containing exactly one vertex of each color class of S and this property holds for every induced subdigraph of D. The ultimate goal in this research area is to obtain a characterization of χ-diperfect digraphs in terms of forbidden induced subdigraphs, but this may be a very difficult problem and not likely to be solved in a near future. Berge showed the first examples of obstructions for χ-diperfect digraphs (i.e. minimal non-χ-diperfect digraphs) by presenting orientations of odd cycles and complements of odd cycles that are not χ-diperfect. In 2022, de Paula Silva, Nunes da Silva and Lee showed characterizations of non-χ-diperfect super-orientations of odd cycles and their complements. Moreover, they showed that these structures are not the only obstructions for χ-diperfect digraphs, by presenting new obstructions with stability number two and three. In this paper, we present new obstructions for χ-diperfect digraphs with arbitrary stability number and arbitrary chromatic number.