(Circular) backbone colouring: forest backbones in planar graphs

Abstract : 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.
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Frédéric Havet, Andrew D. King, Mathieu Liedloff, Ioan Todinca. (Circular) backbone colouring: forest backbones in planar graphs. Discrete Applied Mathematics, Elsevier, 2014, 169, pp.119-134. ⟨10.1016/j.dam.2014.01.011⟩. ⟨hal-00957243⟩

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