(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 {\it $q$-backbone chromatic number} $\BBC_q(G,H)$ is the minimum $k$ such that $G$ can be properly coloured with colours from $\{1, \dots, k\}$, 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 $\BBC_q(G,H)$, depending on the type of the forest (matching, galaxy, spanning tree). Eventually, we discuss a circular version of the problem.