Fixed-parameter complexity of λ-labelings - Fiala, Jiřı́; Kloks, Ton; Kratochvı́l, Jan (Discrete Applied Mathematics, 2001)
A λ-labeling of a graph G is an assignment of labels from the set {0,…, λ} to the vertices of G such that vertices at distance of at most two get different labels and adjacent vertices get labels which are at least two apart. We study the minimum value λ= λ( G) such that G admits a λ-labeling. We show that for every fixed value k⩾4 it is NP-complete to determine whether λ( G)⩽ k. We further investigate this problem for sparse graphs ( k-almost trees), extending the already known result for ordinary trees. In a generalization of this problem we wish to find a labeling such that vertices at distance two are assigned labels that differ by at least q and the labels of adjacent vertices differ by at least p. We denote the minimum λ that allows such a labeling by L( G; p, q). We show several hardness results for L( G; p, q) including that for any p> q⩾1 there is a λ= λ( p, q) such that deciding if L( G; p, q)⩽ λ is NP-complete, and that for p⩾2 q, this decision is NP-complete for every λ⩾ λ( p, q).