(Circular) backbone colouring: Forest backbones in planar graphs

Consider an undirected graph G and a subgraph H of G, on the same vertex set. The q-backbone chromatic number BBCq(G, H) is the minimum k such that G can be properly coloured with colours from {1,...,k}, and moreover for each edge of H, the colours of its ends differ by at least q. In this paper we focus on the case when G is planar and H is a forest. We give a series of NP-hardness results as well as upper bounds for BBCq(G, H), depending on the type of the forest (matching, galaxy, spanning tree). Eventually, we discuss a circular version of the problem. (C) 2014 Elsevier B.V. All rights reserved.