The Multiresolving Sets of Graphs with Prescribed Multisimilar Equivalence Classes
For a set W=w1,w2,…,wk of vertices and a vertex v of a connected graph G, the multirepresentation of v with respect to W is the k-multiset mr(v∣W)=dv,w1,dv,w2,…,dv,wk, where d(v,wi) is the distance between the vertices v and wi for i=1,2,…,k. The set W is a multiresolving set of G if every two disti...
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2018-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2018/8978193 |
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| author | Varanoot Khemmani Supachoke Isariyapalakul |
| author_facet | Varanoot Khemmani Supachoke Isariyapalakul |
| author_sort | Varanoot Khemmani |
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| description | For a set W=w1,w2,…,wk of vertices and a vertex v of a connected graph G, the multirepresentation of v with respect to W is the k-multiset mr(v∣W)=dv,w1,dv,w2,…,dv,wk, where d(v,wi) is the distance between the vertices v and wi for i=1,2,…,k. The set W is a multiresolving set of G if every two distinct vertices of G have distinct multirepresentations with respect to W. The minimum cardinality of a multiresolving set of G is the multidimension dimM(G) of G. It is shown that, for every pair k,n of integers with k≥3 and n≥3(k-1), there is a connected graph G of order n with dimM(G)=k. For a multiset {a1,a2,…,ak} and an integer c, we define {a1,a2,…,ak}+c,c,…,c=a1+c,a2+c,…,ak+c. A multisimilar equivalence relation RW on V(G) with respect to W is defined by u RW v if mr(u∣W)=mrv∣W+cWu,v,cWu,v,…,cWu,v for some integer cW(u,v). 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. |
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| institution | Kabale University |
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| language | English |
| publishDate | 2018-01-01 |
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| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-3d4ec7f050174592bb42330656f5645b2025-02-03T01:25:52ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252018-01-01201810.1155/2018/89781938978193The Multiresolving Sets of Graphs with Prescribed Multisimilar Equivalence ClassesVaranoot Khemmani0Supachoke Isariyapalakul1Department of Mathematics, Srinakharinwirot University, Sukhumvit 23, Bangkok 10110, ThailandDepartment of Mathematics, Srinakharinwirot University, Sukhumvit 23, Bangkok 10110, ThailandFor a set W=w1,w2,…,wk of vertices and a vertex v of a connected graph G, the multirepresentation of v with respect to W is the k-multiset mr(v∣W)=dv,w1,dv,w2,…,dv,wk, where d(v,wi) is the distance between the vertices v and wi for i=1,2,…,k. The set W is a multiresolving set of G if every two distinct vertices of G have distinct multirepresentations with respect to W. The minimum cardinality of a multiresolving set of G is the multidimension dimM(G) of G. It is shown that, for every pair k,n of integers with k≥3 and n≥3(k-1), there is a connected graph G of order n with dimM(G)=k. For a multiset {a1,a2,…,ak} and an integer c, we define {a1,a2,…,ak}+c,c,…,c=a1+c,a2+c,…,ak+c. A multisimilar equivalence relation RW on V(G) with respect to W is defined by u RW v if mr(u∣W)=mrv∣W+cWu,v,cWu,v,…,cWu,v for some integer cW(u,v). 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.http://dx.doi.org/10.1155/2018/8978193 |
| spellingShingle | Varanoot Khemmani Supachoke Isariyapalakul The Multiresolving Sets of Graphs with Prescribed Multisimilar Equivalence Classes International Journal of Mathematics and Mathematical Sciences |
| title | The Multiresolving Sets of Graphs with Prescribed Multisimilar Equivalence Classes |
| title_full | The Multiresolving Sets of Graphs with Prescribed Multisimilar Equivalence Classes |
| title_fullStr | The Multiresolving Sets of Graphs with Prescribed Multisimilar Equivalence Classes |
| title_full_unstemmed | The Multiresolving Sets of Graphs with Prescribed Multisimilar Equivalence Classes |
| title_short | The Multiresolving Sets of Graphs with Prescribed Multisimilar Equivalence Classes |
| title_sort | multiresolving sets of graphs with prescribed multisimilar equivalence classes |
| url | http://dx.doi.org/10.1155/2018/8978193 |
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