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On the Strong Metric Dimension of Tetrahedral Diamond Lattice

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Abstract

A resolving set is a set W of vertices of a connected graph G(V, E) such that for every pair of vertices u, v of G, there exists a vertex \({w\in W}\) with the condition that the length of a shortest path from u to w is different from the length of a shortest path from v to w. A resolving set of minimum cardinality of G is called a metric basis. Metric dimension is the cardinality of a metric basis. A pair of vertices u, v is said to be strongly resolved by a vertex s, if there exists at least one shortest path from s to u passing through v, or a shortest path from s to v passing through u. A set \({W\subseteq V}\), is said to be a strong resolving set if for all pairs \({u,v\notin W }\), there exists some element \({s\in W}\) such that s strongly resolves the pair u, v. A strong resolving set of minimum cardinality is called a strong metric basis. The cardinality of a strong metric basis for G is called the strong metric dimension of G. The strong metric dimension (metric dimension) problem is to find a strong metric basis (metric basis) in the graph. In this paper, we solve the strong metric dimension and the metric dimension problems for the graph of tetrahedral diamond lattice.

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Authors and Affiliations

  1. Department of Information Science, Kuwait University, Safat, Kuwait

    Paul Manuel

  2. School of Electrical Engineering and Computer Science, The University of Newcastle, Callaghan, Australia

    Bharati Rajan

  3. Department of Mathematics, Loyola College, Chennai, India

    Bharati Rajan

  4. School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, Australia

    Cyriac Grigorious & Sudeep Stephen

Authors
  1. Paul Manuel
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  2. Bharati Rajan
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  3. Cyriac Grigorious
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  4. Sudeep Stephen
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Correspondence to Cyriac Grigorious.

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Manuel, P., Rajan, B., Grigorious, C. et al. On the Strong Metric Dimension of Tetrahedral Diamond Lattice. Math.Comput.Sci. 9, 201–208 (2015). https://doi.org/10.1007/s11786-015-0226-0

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  • DOI: https://doi.org/10.1007/s11786-015-0226-0

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