Total Roman 2-Reinforcement of Graphs
A total Roman 2-dominating function (TR2DF) on a graph Γ=V,E is a function l:V⟶0,1,2, satisfying the conditions that (i) for every vertex y∈V with ly=0, either y is adjacent to a vertex labeled 2 under l, or y is adjacent to at least two vertices labeled 1; (ii) the subgraph induced by the set of ve...
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2021-01-01
|
| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2021/5515250 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832561322834788352 |
|---|---|
| author | M. Kheibari H. Abdollahzadeh Ahangar R. Khoeilar S. M. Sheikholeslami |
| author_facet | M. Kheibari H. Abdollahzadeh Ahangar R. Khoeilar S. M. Sheikholeslami |
| author_sort | M. Kheibari |
| collection | DOAJ |
| description | A total Roman 2-dominating function (TR2DF) on a graph Γ=V,E is a function l:V⟶0,1,2, satisfying the conditions that (i) for every vertex y∈V with ly=0, either y is adjacent to a vertex labeled 2 under l, or y is adjacent to at least two vertices labeled 1; (ii) the subgraph induced by the set of vertices with positive weight has no isolated vertex. The weight of a TR2DF l is the value ∑y∈Vly. The total Roman 2-domination number (TR2D-number) of a graph Γ is the minimum weight of a TR2DF on Γ. The total Roman 2-reinforcement number (TR2R-number) of a graph is the minimum number of edges that have to be added to the graph in order to decrease the TR2D-number. In this manuscript, we study the properties of TR2R-number and we present some sharp upper bounds. In particular, we determine the exact value of TR2R-numbers of some classes of graphs. |
| format | Article |
| id | doaj-art-d9b58e3622f64840a6771e781dad7b3b |
| institution | Kabale University |
| issn | 2314-4629 2314-4785 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-d9b58e3622f64840a6771e781dad7b3b2025-02-03T01:25:25ZengWileyJournal of Mathematics2314-46292314-47852021-01-01202110.1155/2021/55152505515250Total Roman 2-Reinforcement of GraphsM. Kheibari0H. Abdollahzadeh Ahangar1R. Khoeilar2S. M. Sheikholeslami3Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, IranDepartment of Mathematics, Babol Noshirvani University of Technology, Babol, IranDepartment of Mathematics, Azarbaijan Shahid Madani University, Tabriz, IranDepartment of Mathematics, Azarbaijan Shahid Madani University, Tabriz, IranA total Roman 2-dominating function (TR2DF) on a graph Γ=V,E is a function l:V⟶0,1,2, satisfying the conditions that (i) for every vertex y∈V with ly=0, either y is adjacent to a vertex labeled 2 under l, or y is adjacent to at least two vertices labeled 1; (ii) the subgraph induced by the set of vertices with positive weight has no isolated vertex. The weight of a TR2DF l is the value ∑y∈Vly. The total Roman 2-domination number (TR2D-number) of a graph Γ is the minimum weight of a TR2DF on Γ. The total Roman 2-reinforcement number (TR2R-number) of a graph is the minimum number of edges that have to be added to the graph in order to decrease the TR2D-number. In this manuscript, we study the properties of TR2R-number and we present some sharp upper bounds. In particular, we determine the exact value of TR2R-numbers of some classes of graphs.http://dx.doi.org/10.1155/2021/5515250 |
| spellingShingle | M. Kheibari H. Abdollahzadeh Ahangar R. Khoeilar S. M. Sheikholeslami Total Roman 2-Reinforcement of Graphs Journal of Mathematics |
| title | Total Roman 2-Reinforcement of Graphs |
| title_full | Total Roman 2-Reinforcement of Graphs |
| title_fullStr | Total Roman 2-Reinforcement of Graphs |
| title_full_unstemmed | Total Roman 2-Reinforcement of Graphs |
| title_short | Total Roman 2-Reinforcement of Graphs |
| title_sort | total roman 2 reinforcement of graphs |
| url | http://dx.doi.org/10.1155/2021/5515250 |
| work_keys_str_mv | AT mkheibari totalroman2reinforcementofgraphs AT habdollahzadehahangar totalroman2reinforcementofgraphs AT rkhoeilar totalroman2reinforcementofgraphs AT smsheikholeslami totalroman2reinforcementofgraphs |