Identifying codes and locating–dominating sets on paths and cycles

• Published in: Discrete Applied Mathematics Vol. 159; no. 15; pp. 1540 - 1547
• Main Authors: Chen, Chunxia, Lu, Changhong, Miao, Zhengke
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 06.09.2011; Elsevier
• Subjects: [formula omitted]-identifying codes; [formula omitted]-locating-dominating sets; Algorithmics. Computability. Computer arithmetics; Applied sciences; Charon; Combinatorial analysis; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Cycles; Exact sciences and technology; Graphs; Information retrieval. Graph; Integers; Mathematical analysis; Mathematics; Paths; Sciences and techniques of general use; Sensors; Shortest-path problems; Theoretical computing; Cycles; r -locating-dominating sets; r -identifying codes; Paths; Integer; r-identifying codes; Dominating set; Computer theory; r-locating-dominating sets; Shortest path; Combinatorics; Optimization
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2011.06.008

Let G = ( V , E ) be a graph and let r ≥ 1 be an integer. For a set D ⊆ V , define N r [ x ] = { y ∈ V : d ( x , y ) ≤ r } and D r ( x ) = N r [ x ] ∩ D , where d ( x , y ) denotes the number of edges in any shortest path between x and y . D is known as an r -identifying code ( r -locating-dominating set, respectively), if for all vertices x ∈ V ( x ∈ V ∖ D , respectively), D r ( x ) are all nonempty and different. Roberts and Roberts [D.L. Roberts, F.S. Roberts, Locating sensors in paths and cycles: the case of 2-identifying codes, European Journal of Combinatorics 29 (2008) 72–82] provided complete results for the paths and cycles when r = 2 . In this paper, we provide results for a remaining open case in cycles and complete results in paths for r -identifying codes; we also give complete results for 2-locating-dominating sets in cycles, which completes the results of Bertrand et al. [N. Bertrand, I. Charon, O. Hudry, A. Lobstein, Identifying and locating–dominating codes on chains and cycles, European Journal of Combinatorics 25 (2004) 969–987]. ► We provide results for a remaining open case in cycles for r -identifying codes. ► We provide complete results in paths for r -identifying codes. ► We give complete results for 2-locating-dominating sets in cycles.