On Mixed Metric Dimension of Some Path Related Graphs
A vertex <inline-formula> <tex-math notation="LaTeX">$k\in V_{G}$ </tex-math></inline-formula> determined two elements (vertices or edges) <inline-formula> <tex-math notation="LaTeX">$\ell,m \in V_{G}\cup E_{G}$ </tex-math></inline-for...
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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/9222153/ |
| _version_ | 1798001917073293312 |
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| author | Hassan Raza Ying Ji Shaojian Qu |
| author_facet | Hassan Raza Ying Ji Shaojian Qu |
| author_sort | Hassan Raza |
| collection | DOAJ |
| description | A vertex <inline-formula> <tex-math notation="LaTeX">$k\in V_{G}$ </tex-math></inline-formula> determined two elements (vertices or edges) <inline-formula> <tex-math notation="LaTeX">$\ell,m \in V_{G}\cup E_{G}$ </tex-math></inline-formula>, if <inline-formula> <tex-math notation="LaTeX">$d_{G}(k,\ell)\neq d_{G}(k,m)$ </tex-math></inline-formula>. A set <inline-formula> <tex-math notation="LaTeX">$R_ {\text {m}}$ </tex-math></inline-formula> of vertices in a graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is a mixed metric generator for <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>, if two distinct elements (vertices or edges) are determined by some vertex set of <inline-formula> <tex-math notation="LaTeX">$R_ {\text {m}}$ </tex-math></inline-formula>. The least number of elements in the vertex set of <inline-formula> <tex-math notation="LaTeX">$R_ {\text {m}}$ </tex-math></inline-formula> is known as mixed metric dimension, and denoted as <inline-formula> <tex-math notation="LaTeX">$dim_{m}(G)$ </tex-math></inline-formula>. In this article, the mixed metric dimension of some path related graphs is obtained. Those path related graphs are <inline-formula> <tex-math notation="LaTeX">$P^{2}_{n}$ </tex-math></inline-formula> the square of a path, <inline-formula> <tex-math notation="LaTeX">$T(P_{n})$ </tex-math></inline-formula> total graph of a path, the middle graph of a path <inline-formula> <tex-math notation="LaTeX">$M(P_{n})$ </tex-math></inline-formula>, and splitting graph of a path <inline-formula> <tex-math notation="LaTeX">$S(P_{n})$ </tex-math></inline-formula>. We proved that these families of graphs have constant and unbounded mixed metric dimension, respectively. We further presented an improved result for the metric dimension of the splitting graph of a path <inline-formula> <tex-math notation="LaTeX">$S(P_{n})$ </tex-math></inline-formula>. |
| first_indexed | 2024-04-11T11:44:56Z |
| format | Article |
| id | doaj.art-e7f5c581aea740edb640d72f9ff6ba79 |
| institution | Directory Open Access Journal |
| issn | 2169-3536 |
| language | English |
| last_indexed | 2024-04-11T11:44:56Z |
| publishDate | 2020-01-01 |
| publisher | IEEE |
| record_format | Article |
| series | IEEE Access |
| spelling | doaj.art-e7f5c581aea740edb640d72f9ff6ba792022-12-22T04:25:42ZengIEEEIEEE Access2169-35362020-01-01818814618815310.1109/ACCESS.2020.30307139222153On Mixed Metric Dimension of Some Path Related GraphsHassan Raza0https://orcid.org/0000-0002-1477-3608Ying Ji1https://orcid.org/0000-0002-2985-6449Shaojian Qu2https://orcid.org/0000-0002-4746-270XBusiness School, University of Shanghai for Science and Technology, Shanghai, ChinaBusiness School, University of Shanghai for Science and Technology, Shanghai, ChinaManagement Engineering School, University of Nanjing for Information Science and Technology, Nanjing, ChinaA vertex <inline-formula> <tex-math notation="LaTeX">$k\in V_{G}$ </tex-math></inline-formula> determined two elements (vertices or edges) <inline-formula> <tex-math notation="LaTeX">$\ell,m \in V_{G}\cup E_{G}$ </tex-math></inline-formula>, if <inline-formula> <tex-math notation="LaTeX">$d_{G}(k,\ell)\neq d_{G}(k,m)$ </tex-math></inline-formula>. A set <inline-formula> <tex-math notation="LaTeX">$R_ {\text {m}}$ </tex-math></inline-formula> of vertices in a graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is a mixed metric generator for <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>, if two distinct elements (vertices or edges) are determined by some vertex set of <inline-formula> <tex-math notation="LaTeX">$R_ {\text {m}}$ </tex-math></inline-formula>. The least number of elements in the vertex set of <inline-formula> <tex-math notation="LaTeX">$R_ {\text {m}}$ </tex-math></inline-formula> is known as mixed metric dimension, and denoted as <inline-formula> <tex-math notation="LaTeX">$dim_{m}(G)$ </tex-math></inline-formula>. In this article, the mixed metric dimension of some path related graphs is obtained. Those path related graphs are <inline-formula> <tex-math notation="LaTeX">$P^{2}_{n}$ </tex-math></inline-formula> the square of a path, <inline-formula> <tex-math notation="LaTeX">$T(P_{n})$ </tex-math></inline-formula> total graph of a path, the middle graph of a path <inline-formula> <tex-math notation="LaTeX">$M(P_{n})$ </tex-math></inline-formula>, and splitting graph of a path <inline-formula> <tex-math notation="LaTeX">$S(P_{n})$ </tex-math></inline-formula>. We proved that these families of graphs have constant and unbounded mixed metric dimension, respectively. We further presented an improved result for the metric dimension of the splitting graph of a path <inline-formula> <tex-math notation="LaTeX">$S(P_{n})$ </tex-math></inline-formula>.https://ieeexplore.ieee.org/document/9222153/Mixed metric dimensionmetric dimensionedge metric dimensionpath related graphs |
| spellingShingle | Hassan Raza Ying Ji Shaojian Qu On Mixed Metric Dimension of Some Path Related Graphs IEEE Access Mixed metric dimension metric dimension edge metric dimension path related graphs |
| title | On Mixed Metric Dimension of Some Path Related Graphs |
| title_full | On Mixed Metric Dimension of Some Path Related Graphs |
| title_fullStr | On Mixed Metric Dimension of Some Path Related Graphs |
| title_full_unstemmed | On Mixed Metric Dimension of Some Path Related Graphs |
| title_short | On Mixed Metric Dimension of Some Path Related Graphs |
| title_sort | on mixed metric dimension of some path related graphs |
| topic | Mixed metric dimension metric dimension edge metric dimension path related graphs |
| url | https://ieeexplore.ieee.org/document/9222153/ |
| work_keys_str_mv | AT hassanraza onmixedmetricdimensionofsomepathrelatedgraphs AT yingji onmixedmetricdimensionofsomepathrelatedgraphs AT shaojianqu onmixedmetricdimensionofsomepathrelatedgraphs |