Identifying codes and locating-dominating sets on paths and cycles

  • Authors:

  • Chunxia Chen;Changhong Lu;Zhengke Miao

  • Affiliations:

  • Department of Mathematics, East China Normal University, Shanghai, 200241, China;Department of Mathematics, East China Normal University, Shanghai, 200241, China;Department of Mathematics, Xuzhou Normal University, Xuzhou, 221116, China

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2011

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Abstract

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].