Fixed-parameter complexity of λ-labelings

Abstract

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⩾2q, this decision is NP-complete for every λ⩾λ(p,q).