On classes of minimal circular-imperfect graphs

Arnaud Pecher 1, 2 Annegret K. Wagler 3
2 Realopt - Reformulations based algorithms for Combinatorial Optimization
LaBRI - Laboratoire Bordelais de Recherche en Informatique, IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : Circular-perfect graphs form a natural superclass of perfect graphs: on the one hand due to their definition by means of a more general coloring concept, on the other hand as an important class of \chi-bound graphs with the smallest non-trivial \chi-binding function \chi(G) ≤ \omega(G) + 1. The Strong Perfect Graph Conjecture, recently settled by Chudnovsky et al. [4], provides a characterization of perfect graphs by means of forbidden subgraphs. It is, therefore, natural to ask for an analogous conjecture for circular-perfect graphs, that is for a characterization of all minimal circular-imperfect graphs. At present, not many minimal circular-imperfect graphs are known. This paper studies the circular-(im)perfection of some families of graphs: normalized circular cliques, partitionable graphs, planar graphs, and complete joins. We thereby exhibit classes of minimal circular-imperfect graphs, namely, certain partitionable webs, a subclass of planar graphs, and odd wheels and odd antiwheels. As those classes appear to be very different from a structural point of view, we infer that formulating an appropriate conjecture for circular-perfect graphs, as analogue to the Strong Perfect Graph Theorem, seems to be difficult.
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Arnaud Pecher, Annegret K. Wagler. On classes of minimal circular-imperfect graphs. Discrete Applied Mathematics, Elsevier, 2008, 156, pp.998--1010. ⟨hal-00307755⟩

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