TY - JOUR

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

AU - Ito, Takehiro

AU - Kawamura, Kazuto

AU - Ono, Hirotaka

AU - Zhou, Xiao

N1 - Funding Information:
We are grateful to Shinya Kumagai for interesting and fruitful discussions. This work is partially supported by JSPS KAKENHI Grant Numbers 25106504 and 25330003 (T. Ito), 24106004 , 25104521 and 26540005 (H. Ono), and 23500001 (X. Zhou).
Publisher Copyright:
© 2014 Elsevier B.V.
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2014

Y1 - 2014

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.

KW - Graph algorithm

KW - L(2, 1)-labeling

KW - PSPACE-complete

KW - Reachability on solution space

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U2 - 10.1016/j.tcs.2014.04.011

DO - 10.1016/j.tcs.2014.04.011

M3 - Article

AN - SCOPUS:84921396446

VL - 544

SP - 84

EP - 97

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - C

ER -