A New Technique to Uniquely Identify the Edges of a Graph
Graphs are useful for analysing the structure models in computer science, operations research, and sociology. The word metric dimension is the basis of the distance function, which has a symmetric property. Moreover, finding the resolving set of a graph is NP-complete, and the possibilities of findi...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-03-01
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| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/15/3/762 |
| _version_ | 1797608826119127040 |
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| author | Hafiz Muhammad Ikhlaq Rashad Ismail Hafiz Muhammad Afzal Siddiqui Muhammad Faisal Nadeem |
| author_facet | Hafiz Muhammad Ikhlaq Rashad Ismail Hafiz Muhammad Afzal Siddiqui Muhammad Faisal Nadeem |
| author_sort | Hafiz Muhammad Ikhlaq |
| collection | DOAJ |
| description | Graphs are useful for analysing the structure models in computer science, operations research, and sociology. The word metric dimension is the basis of the distance function, which has a symmetric property. Moreover, finding the resolving set of a graph is NP-complete, and the possibilities of finding the resolving set are reduced due to the symmetric behaviour of the graph. In this paper, we introduce the idea of the edge-multiset dimension of graphs. A representation of an edge is defined as the multiset of distances between it and the vertices of a set, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mo>⊆</mo><mi>V</mi><mo>(</mo><mi mathvariant="sans-serif">Γ</mi><mo>)</mo></mrow></semantics></math></inline-formula>. If the representation of two different edges is unequal, then <i>B</i> is an edge-multiset resolving a set of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Γ</mi></semantics></math></inline-formula>. The least possible cardinality of the edge-multiset resolving a set is referred to as the edge-multiset dimension of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Γ</mi></semantics></math></inline-formula>. This article presents preliminary results, special conditions, and bounds on the edge-multiset dimension of certain graphs. This research provides new insights into structure models in computer science, operations research, and sociology. They could have implications for developing computer algorithms, aircraft scheduling, and species movement between regions. |
| first_indexed | 2024-03-11T05:50:01Z |
| format | Article |
| id | doaj.art-33dc16bbe1ea4c31bceb58906272f18c |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-11T05:50:01Z |
| publishDate | 2023-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-33dc16bbe1ea4c31bceb58906272f18c2023-11-17T14:10:38ZengMDPI AGSymmetry2073-89942023-03-0115376210.3390/sym15030762A New Technique to Uniquely Identify the Edges of a GraphHafiz Muhammad Ikhlaq0Rashad Ismail1Hafiz Muhammad Afzal Siddiqui2Muhammad Faisal Nadeem3Department of Mathematics, COMSATS University Islamabad (CUI), Lahore Campus, Lahore 54000, PakistanDepartment of Mathematics, Faculty of Science and Arts, King Khalid University, Muhayl Assir 61913, Saudi ArabiaDepartment of Mathematics, COMSATS University Islamabad (CUI), Lahore Campus, Lahore 54000, PakistanDepartment of Mathematics, COMSATS University Islamabad (CUI), Lahore Campus, Lahore 54000, PakistanGraphs are useful for analysing the structure models in computer science, operations research, and sociology. The word metric dimension is the basis of the distance function, which has a symmetric property. Moreover, finding the resolving set of a graph is NP-complete, and the possibilities of finding the resolving set are reduced due to the symmetric behaviour of the graph. In this paper, we introduce the idea of the edge-multiset dimension of graphs. A representation of an edge is defined as the multiset of distances between it and the vertices of a set, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mo>⊆</mo><mi>V</mi><mo>(</mo><mi mathvariant="sans-serif">Γ</mi><mo>)</mo></mrow></semantics></math></inline-formula>. If the representation of two different edges is unequal, then <i>B</i> is an edge-multiset resolving a set of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Γ</mi></semantics></math></inline-formula>. The least possible cardinality of the edge-multiset resolving a set is referred to as the edge-multiset dimension of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Γ</mi></semantics></math></inline-formula>. This article presents preliminary results, special conditions, and bounds on the edge-multiset dimension of certain graphs. This research provides new insights into structure models in computer science, operations research, and sociology. They could have implications for developing computer algorithms, aircraft scheduling, and species movement between regions.https://www.mdpi.com/2073-8994/15/3/762resolving setmetric dimensionedge metric dimensionmixed metric dimensionline graphparaline graph |
| spellingShingle | Hafiz Muhammad Ikhlaq Rashad Ismail Hafiz Muhammad Afzal Siddiqui Muhammad Faisal Nadeem A New Technique to Uniquely Identify the Edges of a Graph Symmetry resolving set metric dimension edge metric dimension mixed metric dimension line graph paraline graph |
| title | A New Technique to Uniquely Identify the Edges of a Graph |
| title_full | A New Technique to Uniquely Identify the Edges of a Graph |
| title_fullStr | A New Technique to Uniquely Identify the Edges of a Graph |
| title_full_unstemmed | A New Technique to Uniquely Identify the Edges of a Graph |
| title_short | A New Technique to Uniquely Identify the Edges of a Graph |
| title_sort | new technique to uniquely identify the edges of a graph |
| topic | resolving set metric dimension edge metric dimension mixed metric dimension line graph paraline graph |
| url | https://www.mdpi.com/2073-8994/15/3/762 |
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