The computational complexity of the backbone coloring problem for planar graphs with connected backbones

In the paper we study the computational complexity of the backbone coloring problem for planar graphs with connected backbones. For every possible value of integer parameters λ≥2λ≥2 and k≥1k≥1 we show that the following problem: Instance:A simple planar graph GG, its connected spanning subgraph (backbone) HH.Question:Is there a λλ-backbone coloring cc of GG with backbone HH such that maxc(V(G))≤kmaxc(V(G))≤k? is either NP-complete or polynomially solvable (by algorithms that run in constant, linear or quadratic time). As a result of these considerations we obtain a complete classification of the computational complexity with respect to the values of λλ and kk.We also study the problem of computing the backbone chromatic number for two special classes of planar graphs: cacti and thorny graphs. We construct an algorithm that runs in O(n3)O(n3) time and solves this problem for cacti and another polynomial algorithm that is 11-absolute approximate for thorny graphs.