On some applications of the selective graph coloring problem

• Selective graph coloring generalizes the usual graph coloring problem.
• Various applications of the selective graph coloring problem are considered.
• Each application is modeled in a different graph class.
• We mention related complexity results and highlight related open questions.

In this paper we present the Selective Graph Coloring Problem, a generalization of the standard graph coloring problem as well as several of its possible applications. Given a graph with a partition of its vertex set into several clusters, we want to select one vertex per cluster such that the chromatic number of the subgraph induced by the selected vertices is minimum. This problem appeared in the literature under different names for specific models and its complexity has recently been studied for different classes of graphs. Here, we describe different models – some already discussed in previous papers and some new ones – in very different contexts under a unified framework based on this graph problem. We point out similarities between these models, offering a new approach to solve them, and show some generic situations where the selective graph coloring problem may be used. We focus on specific graph classes motivated by each model, and we briefly discuss the complexity of the selective graph coloring problem in each one of these graph classes and point out interesting future research directions.