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The Multiresolving Sets of Graphs with Prescribed Multisimilar Equivalence Classes

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  • Varanoot Khemmani
  • Supachoke Isariyapalakul

Abstract

For a set of vertices and a vertex of a connected graph , the multirepresentation of with respect to is the -multiset where is the distance between the vertices and for . The set is a multiresolving set of if every two distinct vertices of have distinct multirepresentations with respect to . The minimum cardinality of a multiresolving set of is the multidimension of . It is shown that, for every pair of integers with and , there is a connected graph of order with . For a multiset and an integer , we define . A multisimilar equivalence relation on with respect to is defined by if for some integer . We study the relationship between the elements in multirepresentations of vertices that belong to the same multisimilar equivalence class and also establish the upper bound for the cardinality of a multisimilar equivalence class. Moreover, a multiresolving set with prescribed multisimilar equivalence classes is presented.

Suggested Citation

  • Varanoot Khemmani & Supachoke Isariyapalakul, 2018. "The Multiresolving Sets of Graphs with Prescribed Multisimilar Equivalence Classes," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2018, pages 1-6, August.
    • Handle: RePEc:hin:jijmms:8978193
    DOI: 10.1155/2018/8978193
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