We study computational complexity of the class of distance-constrained graphlabeling problems from the fixed parameter tractability point of view. Theparameters studied are neighborhood diversity and clique width. We rephrase the distance constrained graph labeling problem as a specificuniform variant of the Channel Assignment problem and show that this problem isfixed parameter tractable when parameterized by the neighborhood diversitytogether with the largest weight. Consequently, every $L(p_1, p_2, \dots,p_k)$-labeling problem is FPT when parameterized by the neighborhood diversity,the maximum $p_i$ and $k.$ Our results yield also FPT algorithms for all $L(p_1, p_2, \dots,p_k)$-labeling problems when parameterized by the size of a minimum vertexcover, answering an open question of Fiala et al.: Parameterized complexity ofcoloring problems: Treewidth versus vertex cover. The same consequence applieson Channel Assignment when the maximum weight is additionally included amongthe parameters. Finally, we show that the uniform variant of the Channel Assignment problembecomes NP-complete when generalized to graphs of bounded clique width.
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