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A Linear Time Algorithm for L(2,1)-Labeling of Trees

  • Toru Hasunuma
  • Toshimasa Ishii
  • Hirotaka Ono
  • Yushi Uno
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5757)

Abstract

An L(2,1)-labeling of a graph G is an assignment f from the vertex set V(G) to the set of nonnegative integers such that |f(x) − f(y)| ≥ 2 if x and y are adjacent and |f(x) − f(y)| ≥ 1 if x and y are at distance 2, for all x and y in V(G). A k-L(2,1)-labeling is an L(2,1)-labeling f:V(G)→{0,...,k}, and the L(2,1)-labeling problem asks the minimum k, which we denote by λ(G), among all possible assignments. It is known that this problem is NP-hard even for graphs of treewidth 2, and tree is one of very few classes for which the problem is polynomially solvable. The running time of the best known algorithm for trees had been O4.5 n) for more than a decade, and an O( min {n 1.751.5 n})-time algorithm has appeared recently, where Δ is the maximum degree of T and n = |V(T)|, however, it has been open if it is solvable in linear time. In this paper, we finally settle this problem for L(2,1)-labeling of trees by establishing a linear time algorithm.

Keywords

Time Algorithm Polynomial Time Algorithm Linear Time Algorithm Input Tree Label Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Toru Hasunuma
    • 1
  • Toshimasa Ishii
    • 2
  • Hirotaka Ono
    • 3
  • Yushi Uno
    • 4
  1. 1.Department of Mathematical and Natural SciencesThe University of TokushimaTokushimaJapan
  2. 2.Department of Information and Management ScienceOtaru University of CommerceOtaruJapan
  3. 3.Department of Computer Science and Communication EngineeringKyushu UniversityFukuokaJapan
  4. 4.Department of Mathematics and Information Sciences, Graduate School of ScienceOsaka Prefecture UniversitySakaiJapan

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