### Abstract

For an integer k ≥ 0, suppose that each vertex v of a graph G has a set C(v) ⊆ {0, 1, . . . , k} of labels, called a list of v. A list L(2, 1)-labeling of G is an assignment of a label in C(v) to each vertex v of G such that every two adjacent vertices receive labels which differ by at least 2 and every two vertices of distance two receive labels which differ by at least 1. In this paper, we study the problem of reconfiguring one list L(2, 1)-labeling of a graph into another list L(2, 1)-labeling of the same graph by changing only one label assignment at a time, while at all times maintaining a list L(2, 1)-labeling. First we show that this decision problem is PSPACE-complete, even for bipartite planar graphs and k ≥ 6. In contrast, we then show that the problem can be solved in linear time for general graphs if k ≤ 4. We finally consider the problem restricted to trees, and give a sufficient condition for which any two list L(2, 1)-labelings of a tree can be transformed into each other.

Original language | English |
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Title of host publication | Algorithms and Computation - 23rd International Symposium, ISAAC 2012, Proceedings |

Pages | 34-43 |

Number of pages | 10 |

Publication status | Published - Dec 31 2012 |

Event | 23rd International Symposium on Algorithms and Computation, ISAAC 2012 - Taipei, Taiwan, Province of China Duration: Dec 19 2012 → Dec 21 2012 |

### 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 | 7676 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 23rd International Symposium on Algorithms and Computation, ISAAC 2012 |
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Country | Taiwan, Province of China |

City | Taipei |

Period | 12/19/12 → 12/21/12 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Algorithms and Computation - 23rd International Symposium, ISAAC 2012, Proceedings*(pp. 34-43). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7676 LNCS).

**Reconfiguration of list L(2, 1)-labelings in a graph.** / Ito, Takehiro; Kawamura, Kazuto; Ono, Hirotaka; Zhou, Xiao.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Algorithms and Computation - 23rd International Symposium, ISAAC 2012, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 7676 LNCS, pp. 34-43, 23rd International Symposium on Algorithms and Computation, ISAAC 2012, Taipei, Taiwan, Province of China, 12/19/12.

}

TY - GEN

T1 - Reconfiguration of list L(2, 1)-labelings in a graph

AU - Ito, Takehiro

AU - Kawamura, Kazuto

AU - Ono, Hirotaka

AU - Zhou, Xiao

PY - 2012/12/31

Y1 - 2012/12/31

N2 - For an integer k ≥ 0, suppose that each vertex v of a graph G has a set C(v) ⊆ {0, 1, . . . , k} of labels, called a list of v. A list L(2, 1)-labeling of G is an assignment of a label in C(v) to each vertex v of G such that every two adjacent vertices receive labels which differ by at least 2 and every two vertices of distance two receive labels which differ by at least 1. In this paper, we study the problem of reconfiguring one list L(2, 1)-labeling of a graph into another list L(2, 1)-labeling of the same graph by changing only one label assignment at a time, while at all times maintaining a list L(2, 1)-labeling. First we show that this decision problem is PSPACE-complete, even for bipartite planar graphs and k ≥ 6. In contrast, we then show that the problem can be solved in linear time for general graphs if k ≤ 4. We finally consider the problem restricted to trees, and give a sufficient condition for which any two list L(2, 1)-labelings of a tree can be transformed into each other.

AB - For an integer k ≥ 0, suppose that each vertex v of a graph G has a set C(v) ⊆ {0, 1, . . . , k} of labels, called a list of v. A list L(2, 1)-labeling of G is an assignment of a label in C(v) to each vertex v of G such that every two adjacent vertices receive labels which differ by at least 2 and every two vertices of distance two receive labels which differ by at least 1. In this paper, we study the problem of reconfiguring one list L(2, 1)-labeling of a graph into another list L(2, 1)-labeling of the same graph by changing only one label assignment at a time, while at all times maintaining a list L(2, 1)-labeling. First we show that this decision problem is PSPACE-complete, even for bipartite planar graphs and k ≥ 6. In contrast, we then show that the problem can be solved in linear time for general graphs if k ≤ 4. We finally consider the problem restricted to trees, and give a sufficient condition for which any two list L(2, 1)-labelings of a tree can be transformed into each other.

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UR - http://www.scopus.com/inward/citedby.url?scp=84871556207&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84871556207

SN - 9783642352607

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 34

EP - 43

BT - Algorithms and Computation - 23rd International Symposium, ISAAC 2012, Proceedings

ER -