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Fixed-parameter complexity of λ-labelings. (English) Zbl 0982.05085
A λ(p,q) labelling of a graph G=(V,E) is an assignment of non-negative integers to the vertices in V, such that adjacent vertices get labels that differ by at least p, and vertices of distance two in G get labels that differ by at least q. L(G;p,q) is the minimum possible maximum value in a λ(p,q) labelling of G. The standard λ-coloring problem is to determine L(G;2,1) of a graph. This paper shows that this problem is solvable in polynomial time on almost k-trees (graphs that can be formed by adding k edges to a tree), and is NP-hard for every fixed maximum label value. It is also shown for all values p>q1 that there are several λ such that deciding if L(G;p,q)λ is NP-complete (taking λ here as fixed part of the problem description). Some other related results are also shown.
MSC:
05C78Graph labelling
68R10Graph theory in connection with computer science (including graph drawing)
05C90Applications of graph theory
05C15Coloring of graphs and hypergraphs