Classes of graphs critical for the edge list-ranking problem
Dmitriy Sergeevich Malyshev · Journal of Applied and Industrial Mathematics · 2014
The edge list-ranking problem is a generalization of the classical edge coloring problem, and it is a mathematical model for some parallel processes. The computational complexity of this problem is under study for graph sets closed under isomorphism and deletion of vertices (hereditary classes). All finitely defined and minor-closed cases are described for which the problem is polynomial-time solvable (unless N=NP).We find the whole set of “critical” graph classes whose inclusion in a finitely defined class is equivalent to intractability of the edge list-ranking problem in this class (unless N=NP). It seems to be the first result on a complete description for nonartificial NP-complete graph problems. For this problem, we prove constructively that, among the inclusion minimal NP-complete hereditary cases, there are exactly five finitely defined classes and the only minor-closed class.