Issue |
RAIRO-Oper. Res.
Volume 57, Number 3, May-June 2023
|
|
---|---|---|
Page(s) | 1195 - 1208 | |
DOI | https://doi.org/10.1051/ro/2023058 | |
Published online | 18 May 2023 |
Graphs with small or large Roman {3}-domination number
1
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, I.R. Iran
2
Department of Mathematics, Babol Noshirvani University of Technology, Shariati Ave., Babol 47148-71167, I.R. Iran
3
LAMDA-RO Laboratory, Department of Mathematics, University of Blida, Blida, Algeria
* Corresponding author: s.m.sheikholeslami@azaruniv.ac.ir; sm_sheikholeslami@yahoo.com
Received:
11
July
2022
Accepted:
23
April
2023
For an integer k ≥ 1, a Roman {k}-dominating function (R{k}DF) on a graph G = (V, E) is a function f : V → {0, 1, …, k} such that for every vertex v ∈ V with f(v) = 0, ∑u∈N(v) f(u) ≥ k, where N(v) is the set of vertices adjacent to v. The weight of an R{k}DF is the sum of its function values over the whole set of vertices, and the Roman {k}-domination number γ{kR}(G) is the minimum weight of an R{k}DF on G. In this paper, we will be focusing on the case k = 3, where trivially for every connected graphs of order n ≥ 3, 3 ≤ γ{kR}(G) ≤n. We characterize all connected graphs G of order n ≥ 3 such that γ{3R}(G) ∈ {3, n − 1, n}, and we improve the previous lower and upper bounds. Moreover, we show that for every tree T of order n ≥ 3, γ{3R}(T) ≥ γ(T) + 2, where γ(T) is the domination number of T, and we characterize the trees attaining this bound.
Mathematics Subject Classification: 05C69
Key words: Roman domination / Roman {k}-domination number
© The authors. Published by EDP Sciences, ROADEF, SMAI 2023
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