Fixed-parameter complexity of λ-labelings

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 sh...

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Bibliographic Details
Published in:Discrete Applied Mathematics Vol. 113; no. 1; pp. 59 - 72
Main Authors: Fiala, Jiřı́, Kloks, Ton, Kratochvı́l, Jan
Format: Journal Article Conference Proceeding
Language:English
Published: Lausanne Elsevier B.V 2001
Amsterdam Elsevier
New York, NY
Subjects:
Algorithmics. Computability. Computer arithmetics
Applied sciences
Channel assignment
Computer science; control theory; systems
Exact sciences and technology
Fixed-parameter complexity
Graph cover
Graph labeling
Theoretical computing
Switzerland
Europe
Graph cover
Channel assignment
Graph labeling
Fixed-parameter complexity
Resource allocation
Channel allocation
Multigraph
NP complete problem
Graph theory
Workshop
Graph labelling
Overlay
1999
Complexity
ISSN:0166-218X, 1872-6771
Online Access:Get full text
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Summary: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).
ISSN:0166-218X
1872-6771
DOI:10.1016/S0166-218X(00)00387-5