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 a graph G such that vertices at distance 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 (where p and q are given positive integers). We denote the minimum number of labels by L(G;p,q) (hence λ(G) = L(G;2, 1)). 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(p,q) is NP-complete.

Research supported in part by the Czech Research Grants GAUK 158/99 and GAČR 201/1996/0194.

Supported by DIMATIA and GAČR 201/99/0242