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Parameterized complexity of distance labeling and uniform channel assignment problems. (English) Zbl 1395.05143
Summary: We rephrase the distance labeling problem as a specific uniform variant of the channel assignment problem and show that the latter one is fixed parameter tractable when parameterized by the neighborhood diversity together with the largest weight. Consequently, the distance labeling problem is $$\mathsf{FPT}$$ when parameterized by the neighborhood diversity, the maximum $$p_i$$ and $$k$$. This is indeed a more general answer to an open question of J. Fiala et al. [Lect. Notes Comput. Sci. 5532, 221–230 (2009; Zbl 1241.68071)].
Finally, we show that the uniform variant of the channel assignment problem becomes $$\mathsf{NP}$$-complete when generalized to graphs of bounded clique width.

##### MSC:
 05C78 Graph labelling (graceful graphs, bandwidth, etc.) 05C82 Small world graphs, complex networks (graph-theoretic aspects) 05C12 Distance in graphs 90B80 Discrete location and assignment
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