On Mixed Metric Dimension of Rotationally Symmetric Graphs
A vertex u ∈ V(G) resolves (distinguish or recognize) two elements (vertices or edges) v, w ∈ E(G)UV(G) if d<sub>G</sub>(u, v) ≠ d<sub>G</sub>(u, w) . A subset L<sub>m</sub> of vertices in a connected graph G is called a mixed metr...
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| Format: | Article |
| Language: | English |
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IEEE
2020-01-01
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| Series: | IEEE Access |
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| Online Access: | https://ieeexplore.ieee.org/document/8937516/ |
| _version_ | 1819173663946047488 |
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| author | Hassan Raza Jia-Bao Liu Shaojian Qu |
| author_facet | Hassan Raza Jia-Bao Liu Shaojian Qu |
| author_sort | Hassan Raza |
| collection | DOAJ |
| description | A vertex u ∈ V(G) resolves (distinguish or recognize) two elements (vertices or edges) v, w ∈ E(G)UV(G) if d<sub>G</sub>(u, v) ≠ d<sub>G</sub>(u, w) . A subset L<sub>m</sub> of vertices in a connected graph G is called a mixed metric generator for G if every two distinct elements (vertices and edges) of G are resolved by some vertex set of L<sub>m</sub>. The minimum cardinality of a mixed metric generator for G is called the mixed metric dimension and is denoted by dim<sub>m</sub>(G). In this paper, we studied the mixed metric dimension for three families of graphs D<sub>n</sub>, A<sub>n</sub>, and R<sub>n</sub>, known from the literature. We proved that, for D<sub>n</sub> the dim<sub>m</sub>(D<sub>n</sub>) = dim<sub>e</sub>(D<sub>n</sub>) = dim(D<sub>n</sub>), when n is even, and for An the dim<sub>m</sub>(A<sub>n</sub>) = dim<sub>e</sub>(A<sub>n</sub>), when n is even and odd. The graph R<sub>n</sub> has mixed metric dimension 5. |
| first_indexed | 2024-12-22T20:26:40Z |
| format | Article |
| id | doaj.art-457b233cd209491ea8059ceb76613c78 |
| institution | Directory Open Access Journal |
| issn | 2169-3536 |
| language | English |
| last_indexed | 2024-12-22T20:26:40Z |
| publishDate | 2020-01-01 |
| publisher | IEEE |
| record_format | Article |
| series | IEEE Access |
| spelling | doaj.art-457b233cd209491ea8059ceb76613c782022-12-21T18:13:43ZengIEEEIEEE Access2169-35362020-01-018115601156910.1109/ACCESS.2019.29611918937516On Mixed Metric Dimension of Rotationally Symmetric GraphsHassan Raza0https://orcid.org/0000-0002-1477-3608Jia-Bao Liu1https://orcid.org/0000-0002-9620-7692Shaojian Qu2https://orcid.org/0000-0002-4746-270XBusiness School, University of Shanghai for Science and Technology, Shanghai, ChinaSchool of Mathematics and Physics, Anhui Jianzhu University, Hefei, ChinaBusiness School, University of Shanghai for Science and Technology, Shanghai, ChinaA vertex u ∈ V(G) resolves (distinguish or recognize) two elements (vertices or edges) v, w ∈ E(G)UV(G) if d<sub>G</sub>(u, v) ≠ d<sub>G</sub>(u, w) . A subset L<sub>m</sub> of vertices in a connected graph G is called a mixed metric generator for G if every two distinct elements (vertices and edges) of G are resolved by some vertex set of L<sub>m</sub>. The minimum cardinality of a mixed metric generator for G is called the mixed metric dimension and is denoted by dim<sub>m</sub>(G). In this paper, we studied the mixed metric dimension for three families of graphs D<sub>n</sub>, A<sub>n</sub>, and R<sub>n</sub>, known from the literature. We proved that, for D<sub>n</sub> the dim<sub>m</sub>(D<sub>n</sub>) = dim<sub>e</sub>(D<sub>n</sub>) = dim(D<sub>n</sub>), when n is even, and for An the dim<sub>m</sub>(A<sub>n</sub>) = dim<sub>e</sub>(A<sub>n</sub>), when n is even and odd. The graph R<sub>n</sub> has mixed metric dimension 5.https://ieeexplore.ieee.org/document/8937516/Mixed metric dimensionmetric dimensionedge metric dimensionrotationally-symmetric |
| spellingShingle | Hassan Raza Jia-Bao Liu Shaojian Qu On Mixed Metric Dimension of Rotationally Symmetric Graphs IEEE Access Mixed metric dimension metric dimension edge metric dimension rotationally-symmetric |
| title | On Mixed Metric Dimension of Rotationally Symmetric Graphs |
| title_full | On Mixed Metric Dimension of Rotationally Symmetric Graphs |
| title_fullStr | On Mixed Metric Dimension of Rotationally Symmetric Graphs |
| title_full_unstemmed | On Mixed Metric Dimension of Rotationally Symmetric Graphs |
| title_short | On Mixed Metric Dimension of Rotationally Symmetric Graphs |
| title_sort | on mixed metric dimension of rotationally symmetric graphs |
| topic | Mixed metric dimension metric dimension edge metric dimension rotationally-symmetric |
| url | https://ieeexplore.ieee.org/document/8937516/ |
| work_keys_str_mv | AT hassanraza onmixedmetricdimensionofrotationallysymmetricgraphs AT jiabaoliu onmixedmetricdimensionofrotationallysymmetricgraphs AT shaojianqu onmixedmetricdimensionofrotationallysymmetricgraphs |