Dominant mixed metric dimension of graph
For k−ordered set W = {s1, s2, …, sk} of vertex set G, the representation of a vertex or edge a of G with respect to W is r(a|W) = (d(a, s1), d(a, s2), …, d(a, sk)) where a is vertex so that d(a, si) is a distance between the vertex a and the vertices in W and a = uv is edge so that d(a, si) = min{d...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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International Academic Press
2024
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| Online Access: | http://psasir.upm.edu.my/id/eprint/117899/ http://psasir.upm.edu.my/id/eprint/117899/1/117899.pdf |
| _version_ | 1848867374030127104 |
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| author | Alfarisi, Ridho Husain, Sharifah Kartini Said Susilowati, Liliek Kristiana, Arika Indah |
| author_facet | Alfarisi, Ridho Husain, Sharifah Kartini Said Susilowati, Liliek Kristiana, Arika Indah |
| author_sort | Alfarisi, Ridho |
| building | UPM Institutional Repository |
| collection | Online Access |
| description | For k−ordered set W = {s1, s2, …, sk} of vertex set G, the representation of a vertex or edge a of G with respect to W is r(a|W) = (d(a, s1), d(a, s2), …, d(a, sk)) where a is vertex so that d(a, si) is a distance between the vertex a and the vertices in W and a = uv is edge so that d(a, si) = min{d(u, si), d(v, si)}. The set W is a mixed resolving set of G if r(a|W) ≠ r(b|W) for every pair a, b of distinct vertices or edge of G. The minimum mixed resolving set W is a mixed basis of G. If G has a mixed basis, then its cardinality is called a mixed metric dimension, denoted by dimm(G). A set W of vertices in G is a dominating set for G if every vertex of G that is not in W is adjacent to some vertex of W. The minimum cardinality of the dominant set is the domination number, denoted by γ(G). A vertex set of some vertices in G that is both mixed resolving and dominating set is a mixed resolving dominating set. The minimum cardinality of the dominant set with mixed resolving is called the dominant mixed metric dimension, denoted by γmr(G). In our paper, we investigate the establishment of sharp bounds of the dominant mixed metric dimension of G and determine the exact value of some family graphs. |
| first_indexed | 2025-11-15T14:35:28Z |
| format | Article |
| id | upm-117899 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-15T14:35:28Z |
| publishDate | 2024 |
| publisher | International Academic Press |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-1178992025-06-16T07:47:40Z http://psasir.upm.edu.my/id/eprint/117899/ Dominant mixed metric dimension of graph Alfarisi, Ridho Husain, Sharifah Kartini Said Susilowati, Liliek Kristiana, Arika Indah For k−ordered set W = {s1, s2, …, sk} of vertex set G, the representation of a vertex or edge a of G with respect to W is r(a|W) = (d(a, s1), d(a, s2), …, d(a, sk)) where a is vertex so that d(a, si) is a distance between the vertex a and the vertices in W and a = uv is edge so that d(a, si) = min{d(u, si), d(v, si)}. The set W is a mixed resolving set of G if r(a|W) ≠ r(b|W) for every pair a, b of distinct vertices or edge of G. The minimum mixed resolving set W is a mixed basis of G. If G has a mixed basis, then its cardinality is called a mixed metric dimension, denoted by dimm(G). A set W of vertices in G is a dominating set for G if every vertex of G that is not in W is adjacent to some vertex of W. The minimum cardinality of the dominant set is the domination number, denoted by γ(G). A vertex set of some vertices in G that is both mixed resolving and dominating set is a mixed resolving dominating set. The minimum cardinality of the dominant set with mixed resolving is called the dominant mixed metric dimension, denoted by γmr(G). In our paper, we investigate the establishment of sharp bounds of the dominant mixed metric dimension of G and determine the exact value of some family graphs. International Academic Press 2024 Article PeerReviewed text en http://psasir.upm.edu.my/id/eprint/117899/1/117899.pdf Alfarisi, Ridho and Husain, Sharifah Kartini Said and Susilowati, Liliek and Kristiana, Arika Indah (2024) Dominant mixed metric dimension of graph. Statistics, Optimization & Information Computing, 12 (6). pp. 1826-1833. ISSN 2310-5070; eISSN: 2311-004X http://www.iapress.org/index.php/soic/article/view/1925 10.19139/soic-2310-5070-1925 |
| spellingShingle | Alfarisi, Ridho Husain, Sharifah Kartini Said Susilowati, Liliek Kristiana, Arika Indah Dominant mixed metric dimension of graph |
| title | Dominant mixed metric dimension of graph |
| title_full | Dominant mixed metric dimension of graph |
| title_fullStr | Dominant mixed metric dimension of graph |
| title_full_unstemmed | Dominant mixed metric dimension of graph |
| title_short | Dominant mixed metric dimension of graph |
| title_sort | dominant mixed metric dimension of graph |
| url | http://psasir.upm.edu.my/id/eprint/117899/ http://psasir.upm.edu.my/id/eprint/117899/ http://psasir.upm.edu.my/id/eprint/117899/ http://psasir.upm.edu.my/id/eprint/117899/1/117899.pdf |