We consider the problem of coloring a planar graph with the minimum number of colors such that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem. We present a complete picture for the case with a single forbidden connected (induced or non-induced) subgraph. The 2-coloring problem is NP-hard if the forbidden subgraph is a tree with at least two edges, and it is polynomially solvable in all other cases. The 3-coloring problem is NP-hard if the forbidden subgraph is a path, and it is polynomially solvable in all other cases. We also derive results for several forbidden sets of cycles.
CITATION STYLE
Broersma, H., Fomin, F. V., Kratochvíl, J., & Woeginger, G. J. (2002). Planar graph coloring with forbidden subgraphs: Why trees and paths are dangerous. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 2368, pp. 160–169). Springer Verlag. https://doi.org/10.1007/3-540-45471-3_17
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