### 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 assignment 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. Tree is one of a few classes for which the problem is polynomially solvable, but still only an time algorithm for a tree T has been known so far, where Δ is the maximum degree of T and n∈=∈|V(T)|. In this paper, we first show that an existent necessary condition for λ(T)∈=∈Δ∈+∈1 is also sufficient for a tree T with , which leads a linear time algorithm for computing λ(T) under this condition. We then show that λ(T) can be computed in time for any tree T. Combining these, we finally obtain an time algorithm, which substantially improves upon previously known results.

Original language | English |
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Title of host publication | Algorithm Theory - SWAT 2008 - 11th Scandinavian Workshop on Algorithm Theory, Proceedings |

Pages | 185-197 |

Number of pages | 13 |

DOIs | |

Publication status | Published - Oct 27 2008 |

Event | 11th Scandinavian Workshop on Algorithm Theory, SWAT 2008 - Gothenburg, Sweden Duration: Jul 2 2008 → Jul 4 2008 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 5124 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 11th Scandinavian Workshop on Algorithm Theory, SWAT 2008 |
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Country | Sweden |

City | Gothenburg |

Period | 7/2/08 → 7/4/08 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

^{1.75}) algorithm for L(2,1)-labeling of trees. In

*Algorithm Theory - SWAT 2008 - 11th Scandinavian Workshop on Algorithm Theory, Proceedings*(pp. 185-197). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5124 LNCS). https://doi.org/10.1007/978-3-540-69903-3_18