On The Total Roman Domination in Trees
A total Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the following conditions: (i) every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2 and (ii) the subgraph of G induced by the set of all vertices of positive weight has n...
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| Format: | Article |
| Language: | English |
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University of Zielona Góra
2019-05-01
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| Series: | Discussiones Mathematicae Graph Theory |
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| Online Access: | https://doi.org/10.7151/dmgt.2099 |
| _version_ | 1827844621773832192 |
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| author | Amjadi Jafar Sheikholeslami Seyed Mahmoud Soroudi Marzieh |
| author_facet | Amjadi Jafar Sheikholeslami Seyed Mahmoud Soroudi Marzieh |
| author_sort | Amjadi Jafar |
| collection | DOAJ |
| description | A total Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the following conditions: (i) every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2 and (ii) the subgraph of G induced by the set of all vertices of positive weight has no isolated vertex. The weight of a total Roman dominating function f is the value f(V (G)) = Σu∈V(G)f (u). The total Roman domination number γtR(G) is the minimum weight of a total Roman dominating function of G. Ahangar et al. in [H.A. Ahangar, M.A. Henning, V. Samodivkin and I.G. Yero, Total Roman domination in graphs, Appl. Anal. Discrete Math. 10 (2016) 501–517] recently showed that for any graph G without isolated vertices, 2γ(G) ≤ γtR(G) ≤ 3γ(G), where γ(G) is the domination number of G, and they raised the problem of characterizing the graphs G achieving these upper and lower bounds. In this paper, we provide a constructive characterization of these trees. |
| first_indexed | 2024-03-12T08:44:31Z |
| format | Article |
| id | doaj.art-0e705819c0584e1cb28f30a541d02638 |
| institution | Directory Open Access Journal |
| issn | 2083-5892 |
| language | English |
| last_indexed | 2024-03-12T08:44:31Z |
| publishDate | 2019-05-01 |
| publisher | University of Zielona Góra |
| record_format | Article |
| series | Discussiones Mathematicae Graph Theory |
| spelling | doaj.art-0e705819c0584e1cb28f30a541d026382023-09-02T16:29:33ZengUniversity of Zielona GóraDiscussiones Mathematicae Graph Theory2083-58922019-05-0139251953210.7151/dmgt.2099dmgt.2099On The Total Roman Domination in TreesAmjadi Jafar0Sheikholeslami Seyed Mahmoud1Soroudi Marzieh2Department of Mathematics, Azarbaijan Shahid Madani University,Tabriz, I.R. IranDepartment of Mathematics, Azarbaijan Shahid Madani University,Tabriz, I.R. IranDepartment of Mathematics, Azarbaijan Shahid Madani University,Tabriz, I.R. IranA total Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the following conditions: (i) every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2 and (ii) the subgraph of G induced by the set of all vertices of positive weight has no isolated vertex. The weight of a total Roman dominating function f is the value f(V (G)) = Σu∈V(G)f (u). The total Roman domination number γtR(G) is the minimum weight of a total Roman dominating function of G. Ahangar et al. in [H.A. Ahangar, M.A. Henning, V. Samodivkin and I.G. Yero, Total Roman domination in graphs, Appl. Anal. Discrete Math. 10 (2016) 501–517] recently showed that for any graph G without isolated vertices, 2γ(G) ≤ γtR(G) ≤ 3γ(G), where γ(G) is the domination number of G, and they raised the problem of characterizing the graphs G achieving these upper and lower bounds. In this paper, we provide a constructive characterization of these trees.https://doi.org/10.7151/dmgt.2099total roman dominating functiontotal roman domination numbertrees05c69 |
| spellingShingle | Amjadi Jafar Sheikholeslami Seyed Mahmoud Soroudi Marzieh On The Total Roman Domination in Trees Discussiones Mathematicae Graph Theory total roman dominating function total roman domination number trees 05c69 |
| title | On The Total Roman Domination in Trees |
| title_full | On The Total Roman Domination in Trees |
| title_fullStr | On The Total Roman Domination in Trees |
| title_full_unstemmed | On The Total Roman Domination in Trees |
| title_short | On The Total Roman Domination in Trees |
| title_sort | on the total roman domination in trees |
| topic | total roman dominating function total roman domination number trees 05c69 |
| url | https://doi.org/10.7151/dmgt.2099 |
| work_keys_str_mv | AT amjadijafar onthetotalromandominationintrees AT sheikholeslamiseyedmahmoud onthetotalromandominationintrees AT soroudimarzieh onthetotalromandominationintrees |