Strong Total Monophonic Problems in Product Graphs, Networks, and Its Computational Complexity
Let G be a graph with vertex set as VG and edge set as EG which is simple as well as connected. The problem of strong total monophonic set is to find the set of vertices T⊆VG, which contains no isolated vertices, and all the vertices in VG\T lie on a fixed unique chordless path between the pair of v...
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| Published in: | Journal of Mathematics Vol. 2022; no. 1 |
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| Main Authors: | , , , , , |
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| Language: | English |
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Hindawi
2022
John Wiley & Sons, Inc Wiley |
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| ISSN: | 2314-4629, 2314-4785 |
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| Abstract | Let G be a graph with vertex set as VG and edge set as EG which is simple as well as connected. The problem of strong total monophonic set is to find the set of vertices T⊆VG, which contains no isolated vertices, and all the vertices in VG\T lie on a fixed unique chordless path between the pair of vertices in T. The cardinality of strong total monophonic set which is minimum is defined as strong total monophonic number, denoted by smtG. We proved the NP-completeness of strong total monophonic set for general graphs. The strong total monophonic number of certain graphs and networks is derived. If l,m,n are positive integers with 5≤l≤m≤n and m≤2l−1, then we can construct a connected graph G with strong monophonic number l and strong total monophonic number m. |
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| AbstractList | Let G be a graph with vertex set as VG and edge set as EG which is simple as well as connected. The problem of strong total monophonic set is to find the set of vertices T⊆VG, which contains no isolated vertices, and all the vertices in VG\T lie on a fixed unique chordless path between the pair of vertices in T. The cardinality of strong total monophonic set which is minimum is defined as strong total monophonic number, denoted by smtG. We proved the NP-completeness of strong total monophonic set for general graphs. The strong total monophonic number of certain graphs and networks is derived. If l,m,n are positive integers with 5≤l≤m≤n and m≤2l−1, then we can construct a connected graph G with strong monophonic number l and strong total monophonic number m. Let G be a graph with vertex set as V ( G ) and edge set as E ( G ) which is simple as well as connected. The problem of strong total monophonic set is to find the set of vertices T ⊆ V ( G ), which contains no isolated vertices, and all the vertices in V ( G )\ T lie on a fixed unique chordless path between the pair of vertices in T . The cardinality of strong total monophonic set which is minimum is defined as strong total monophonic number, denoted by s m t ( G ). We proved the NP‐completeness of strong total monophonic set for general graphs. The strong total monophonic number of certain graphs and networks is derived. If l , m , n are positive integers with 5 ≤ l ≤ m ≤ n and m ≤ 2 l − 1, then we can construct a connected graph G with strong monophonic number l and strong total monophonic number m . Let G be a graph with vertex set as V(G) and edge set as E(G) which is simple as well as connected. The problem of strong total monophonic set is to find the set of vertices T⊆V(G), which contains no isolated vertices, and all the vertices in V(G) lie on a fixed unique chordless path between the pair of vertices in T. The cardinality of strong total monophonic set which is minimum is defined as strong total monophonic number, denoted by sm[sub.t](G). We proved the NP-completeness of strong total monophonic set for general graphs. The strong total monophonic number of certain graphs and networks is derived. If l,m,n are positive integers with 5≤l≤m≤n and m≤2l-1, then we can construct a connected graph G with strong monophonic number l and strong total monophonic number m. |
| Audience | Academic |
| Author | Amirtha Raja, S. Arul Mathew, Deepa Varghese, Eddith Sarah Xavier, D. Antony Ahmed, Hanan Alsinai, Ammar |
| Author_xml | – sequence: 1 givenname: Eddith Sarah orcidid: 0000-0001-8236-5756 surname: Varghese fullname: Varghese, Eddith Sarah organization: Department of MathematicsLoyola College (Affliated to the University of Madras)ChennaiIndia – sequence: 2 givenname: D. Antony orcidid: 0000-0002-3808-8592 surname: Xavier fullname: Xavier, D. Antony organization: Department of MathematicsLoyola College (Affliated to the University of Madras)ChennaiIndia – sequence: 3 givenname: Ammar orcidid: 0000-0002-5221-0574 surname: Alsinai fullname: Alsinai, Ammar organization: Department of Studies in MathematicsUniversity of MysoreManasagangothriMysuru - 570 006KarnatakaIndiauni-mysore.ac.in – sequence: 4 givenname: Deepa orcidid: 0000-0002-9410-8834 surname: Mathew fullname: Mathew, Deepa organization: Department of MathematicsSt. Joseph’s CollegeBangaloreIndia – sequence: 5 givenname: S. Arul orcidid: 0000-0001-5920-0726 surname: Amirtha Raja fullname: Amirtha Raja, S. Arul organization: Department of MathematicsSt. Joseph’s College of EngineeringChennaiIndiastjosephs.ac.in – sequence: 6 givenname: Hanan orcidid: 0000-0002-4008-4873 surname: Ahmed fullname: Ahmed, Hanan organization: Department of MathematicsIbb UniversityIbbYemenibb-univ.net |
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| Cites_doi | 10.1137/0211015 10.12988/imf.2007.07153 10.2298/fil1206081j 10.1007/s40840-019-00833-6 10.35940/ijeat.a1231.109119 10.5539/jmr.v11n2p20 10.1016/j.disc.2009.09.018 10.1016/0895-7177(93)90259-2 10.7151/dmgt.2139 10.1016/j.disc.2006.08.002 10.1016/j.dam.2009.11.016 |
| ContentType | Journal Article |
| Copyright | Copyright © 2022 Eddith Sarah Varghese et al. COPYRIGHT 2022 John Wiley & Sons, Inc. Copyright © 2022 Eddith Sarah Varghese et al. This is an open access article distributed under the Creative Commons Attribution License (the “License”), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. https://creativecommons.org/licenses/by/4.0 |
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| References | e_1_2_8_16_2 e_1_2_8_17_2 e_1_2_8_19_2 e_1_2_8_13_2 West D. B. (e_1_2_8_1_2) 2001 e_1_2_8_15_2 Sudhahar P. A. P. (e_1_2_8_9_2) 2018; 13 e_1_2_8_2_2 e_1_2_8_4_2 e_1_2_8_3_2 e_1_2_8_6_2 e_1_2_8_5_2 Santhakumaran A. P. (e_1_2_8_8_2) 2014; 9 e_1_2_8_7_2 Ahangar H. A. (e_1_2_8_12_2) 2016; 100 Varghese E. S. (e_1_2_8_10_2) 2021; 12 Bermudo S. (e_1_2_8_14_2) 2010; 96 e_1_2_8_11_2 West D. B. (e_1_2_8_18_2) 2001 |
| References_xml | – volume-title: Introduction to Graph Theory year: 2001 ident: e_1_2_8_18_2 – volume: 100 start-page: 253 year: 2016 ident: e_1_2_8_12_2 article-title: The total geodetic number of a graph publication-title: Utilitas Mathematica – volume: 12 start-page: 509 year: 2021 ident: e_1_2_8_10_2 article-title: Strong open monophonic number of a graph publication-title: Communications in Mathematics and Applications – ident: e_1_2_8_19_2 – ident: e_1_2_8_2_2 doi: 10.1137/0211015 – ident: e_1_2_8_4_2 doi: 10.12988/imf.2007.07153 – ident: e_1_2_8_6_2 doi: 10.2298/fil1206081j – ident: e_1_2_8_17_2 doi: 10.1007/s40840-019-00833-6 – ident: e_1_2_8_11_2 doi: 10.35940/ijeat.a1231.109119 – ident: e_1_2_8_16_2 doi: 10.5539/jmr.v11n2p20 – volume: 9 year: 2014 ident: e_1_2_8_8_2 article-title: The total open monophonic number of a graph publication-title: Journal of Advances in Mathematics – volume-title: Introduction to Graph Theory year: 2001 ident: e_1_2_8_1_2 – ident: e_1_2_8_15_2 doi: 10.1016/j.disc.2009.09.018 – volume: 13 year: 2018 ident: e_1_2_8_9_2 article-title: The connected total monophonic domination number of a graph publication-title: International Journal of Applied Engineering Research – ident: e_1_2_8_3_2 doi: 10.1016/0895-7177(93)90259-2 – volume: 96 start-page: 469 year: 2010 ident: e_1_2_8_14_2 article-title: On geodetic and k-geodetic sets in graphs publication-title: Ars Combinatoria-Waterloo then Winnipeg – ident: e_1_2_8_5_2 doi: 10.7151/dmgt.2139 – ident: e_1_2_8_7_2 doi: 10.1016/j.disc.2006.08.002 – ident: e_1_2_8_13_2 doi: 10.1016/j.dam.2009.11.016 |
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| Snippet | Let G be a graph with vertex set as VG and edge set as EG which is simple as well as connected. The problem of strong total monophonic set is to find the set... Let G be a graph with vertex set as V ( G ) and edge set as E ( G ) which is simple as well as connected. The problem of strong total monophonic set is to find... Let G be a graph with vertex set as V(G) and edge set as E(G) which is simple as well as connected. The problem of strong total monophonic set is to find the... |
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| Title | Strong Total Monophonic Problems in Product Graphs, Networks, and Its Computational Complexity |
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