Bounds on the Double Italian Domination Number of a Graph
For a graph G, a Roman {3}-dominating function is a function f : V → {0, 1, 2, 3} having the property that for every vertex u ∈ V, if f(u) ∈ {0, 1}, then f(N[u]) ≥ 3. The weight of a Roman {3}-dominating function is the sum w(f) = f(V) = Σv∈V f(v), and the minimum weight of a Roman {3}-dominating fu...
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| Format: | Article |
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University of Zielona Góra
2022-11-01
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| Series: | Discussiones Mathematicae Graph Theory |
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| Online Access: | https://doi.org/10.7151/dmgt.2330 |
| _version_ | 1797717964304154624 |
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| author | Azvin Farzaneh Rad Nader Jafari |
| author_facet | Azvin Farzaneh Rad Nader Jafari |
| author_sort | Azvin Farzaneh |
| collection | DOAJ |
| description | For a graph G, a Roman {3}-dominating function is a function f : V → {0, 1, 2, 3} having the property that for every vertex u ∈ V, if f(u) ∈ {0, 1}, then f(N[u]) ≥ 3. The weight of a Roman {3}-dominating function is the sum w(f) = f(V) = Σv∈V f(v), and the minimum weight of a Roman {3}-dominating function is the Roman {3}-domination number, denoted by γ{R3}(G). In this paper, we present a sharp lower bound for the double Italian domination number of a graph, and improve previous bounds given in [D.A. Mojdeh and L. Volkmann, Roman {3}-domination (double Italian domination), Discrete Appl. Math. 283 (2022) 555–564]. We also present a probabilistic upper bound for a generalized version of double Italian domination number of a graph, and show that the given bound is asymptotically best possible. |
| first_indexed | 2024-03-12T08:44:04Z |
| format | Article |
| id | doaj.art-9c68c6dc40a544c486872a6437689965 |
| institution | Directory Open Access Journal |
| issn | 2083-5892 |
| language | English |
| last_indexed | 2024-03-12T08:44:04Z |
| publishDate | 2022-11-01 |
| publisher | University of Zielona Góra |
| record_format | Article |
| series | Discussiones Mathematicae Graph Theory |
| spelling | doaj.art-9c68c6dc40a544c486872a64376899652023-09-02T16:32:48ZengUniversity of Zielona GóraDiscussiones Mathematicae Graph Theory2083-58922022-11-014241129113710.7151/dmgt.2330Bounds on the Double Italian Domination Number of a GraphAzvin Farzaneh0Rad Nader Jafari1Department of Mathematics, Shahed University, Tehran, IranDepartment of Mathematics, Shahed University, Tehran, IranFor a graph G, a Roman {3}-dominating function is a function f : V → {0, 1, 2, 3} having the property that for every vertex u ∈ V, if f(u) ∈ {0, 1}, then f(N[u]) ≥ 3. The weight of a Roman {3}-dominating function is the sum w(f) = f(V) = Σv∈V f(v), and the minimum weight of a Roman {3}-dominating function is the Roman {3}-domination number, denoted by γ{R3}(G). In this paper, we present a sharp lower bound for the double Italian domination number of a graph, and improve previous bounds given in [D.A. Mojdeh and L. Volkmann, Roman {3}-domination (double Italian domination), Discrete Appl. Math. 283 (2022) 555–564]. We also present a probabilistic upper bound for a generalized version of double Italian domination number of a graph, and show that the given bound is asymptotically best possible.https://doi.org/10.7151/dmgt.2330italian dominationdouble italian dominationprobabilistic methods05c69 |
| spellingShingle | Azvin Farzaneh Rad Nader Jafari Bounds on the Double Italian Domination Number of a Graph Discussiones Mathematicae Graph Theory italian domination double italian domination probabilistic methods 05c69 |
| title | Bounds on the Double Italian Domination Number of a Graph |
| title_full | Bounds on the Double Italian Domination Number of a Graph |
| title_fullStr | Bounds on the Double Italian Domination Number of a Graph |
| title_full_unstemmed | Bounds on the Double Italian Domination Number of a Graph |
| title_short | Bounds on the Double Italian Domination Number of a Graph |
| title_sort | bounds on the double italian domination number of a graph |
| topic | italian domination double italian domination probabilistic methods 05c69 |
| url | https://doi.org/10.7151/dmgt.2330 |
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