Graphs with mixed metric dimension three and related algorithms
Let $ G = (V, E) $ be a simple connected graph. A vertex $ x\in V(G) $ resolves the elements $ u, v\in E(G)\cup V(G) $ if $ d_G(x, u)\neq d_G(x, v) $. A subset $ S\subseteq V(G) $ is a mixed metric resolving set for $ G $ if every two elements of $ G $ are resolved by some vertex of $ S $. A set of...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-05-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2023854?viewType=HTML |
| _version_ | 1797816992871219200 |
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| author | Dalal Awadh Alrowaili Uzma Ahmad Saira Hameeed Muhammad Javaid |
| author_facet | Dalal Awadh Alrowaili Uzma Ahmad Saira Hameeed Muhammad Javaid |
| author_sort | Dalal Awadh Alrowaili |
| collection | DOAJ |
| description | Let $ G = (V, E) $ be a simple connected graph. A vertex $ x\in V(G) $ resolves the elements $ u, v\in E(G)\cup V(G) $ if $ d_G(x, u)\neq d_G(x, v) $. A subset $ S\subseteq V(G) $ is a mixed metric resolving set for $ G $ if every two elements of $ G $ are resolved by some vertex of $ S $. A set of smallest cardinality of mixed metric generator for $ G $ is called the mixed metric dimension. In this paper trees and unicyclic graphs having mixed dimension three are classified. The main aim is to investigate the structure of a simple connected graph having mixed dimension three with respect to the order of graph, maximum degree of basis elements and distance partite sets of basis elements. In particular to find necessary and sufficient conditions for a graph to have mixed metric dimension 3. Moreover three separate algorithms are developed for trees, unicyclic graphs and in general for simple connected graph $ J_{n}\ncong P_{n} $ with $ n\geq 3 $ to determine "whether these graphs have mixed dimension three or not?". If these graphs have mixed dimension three, then these algorithms provide a mixed basis of an input graph. |
| first_indexed | 2024-03-13T08:46:46Z |
| format | Article |
| id | doaj.art-24d5b25c290e4a24915b4b71700080ff |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-03-13T08:46:46Z |
| publishDate | 2023-05-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-24d5b25c290e4a24915b4b71700080ff2023-05-30T01:35:02ZengAIMS PressAIMS Mathematics2473-69882023-05-0187167081672310.3934/math.2023854Graphs with mixed metric dimension three and related algorithmsDalal Awadh Alrowaili 0Uzma Ahmad 1Saira Hameeed2Muhammad Javaid 31. Mathematics Department, College of Science, Jouf University, P.O. Box: 2014, Sakaka, Saudi Arabia2. Department of Mathematics, University of the Punjab, Lahore, Pakistan2. Department of Mathematics, University of the Punjab, Lahore, Pakistan3. Department of Mathematics, School of Science, University of Management and Technology, Lahore 54770, PakistanLet $ G = (V, E) $ be a simple connected graph. A vertex $ x\in V(G) $ resolves the elements $ u, v\in E(G)\cup V(G) $ if $ d_G(x, u)\neq d_G(x, v) $. A subset $ S\subseteq V(G) $ is a mixed metric resolving set for $ G $ if every two elements of $ G $ are resolved by some vertex of $ S $. A set of smallest cardinality of mixed metric generator for $ G $ is called the mixed metric dimension. In this paper trees and unicyclic graphs having mixed dimension three are classified. The main aim is to investigate the structure of a simple connected graph having mixed dimension three with respect to the order of graph, maximum degree of basis elements and distance partite sets of basis elements. In particular to find necessary and sufficient conditions for a graph to have mixed metric dimension 3. Moreover three separate algorithms are developed for trees, unicyclic graphs and in general for simple connected graph $ J_{n}\ncong P_{n} $ with $ n\geq 3 $ to determine "whether these graphs have mixed dimension three or not?". If these graphs have mixed dimension three, then these algorithms provide a mixed basis of an input graph.https://www.aimspress.com/article/doi/10.3934/math.2023854?viewType=HTMLmetric dimensionfractional metric dimensionmodified prism networks |
| spellingShingle | Dalal Awadh Alrowaili Uzma Ahmad Saira Hameeed Muhammad Javaid Graphs with mixed metric dimension three and related algorithms AIMS Mathematics metric dimension fractional metric dimension modified prism networks |
| title | Graphs with mixed metric dimension three and related algorithms |
| title_full | Graphs with mixed metric dimension three and related algorithms |
| title_fullStr | Graphs with mixed metric dimension three and related algorithms |
| title_full_unstemmed | Graphs with mixed metric dimension three and related algorithms |
| title_short | Graphs with mixed metric dimension three and related algorithms |
| title_sort | graphs with mixed metric dimension three and related algorithms |
| topic | metric dimension fractional metric dimension modified prism networks |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023854?viewType=HTML |
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