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Peer-reviewedCC BY 4.0Mathematics

Graphs with Large Vertex–Edge Roman {2}-Domination Number

  • Ahlam Almulhim1
  1. 1Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa 31982, Saudi Arabia
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Abstract

For a graph G=(V,E), a function f:V⟶{0,1,2} is a vertex–edge Roman {2}-dominating function (veR2DF) if for every edge xy∈E, either f(x)+f(y)≥1 or ∑z∈N(x)∪N(y)f(z)≥2. The sum ∑v∈Vf(v) is the weight of the function f. The vertex–edge Roman {2}-domination number of G, denoted by γveR2(G), is the minimum weight of a veR2DF of G. In this paper, we prove that for any connected graph G of order n≥4, then γveR2(G)≤n2, and this result is sharp. We also prove Nordhaus–Gaddum inequalities for γveR2(G). In addition, we characterize trees with equal vertex–edge Roman {2}-domination, vertex–edge domination, and domination numbers.

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Cite

BibTeXRIS
BibTeX
@article{almulhim2026graphs,
  title = {{Graphs with Large Vertex–Edge Roman 2-Domination Number}},
  author = {Ahlam Almulhim},
  journal = {Mathematics},
  year = {2026},
  month = oct,
  doi = {10.3390/math14203819},
  url = {https://www.mdpi.com/2227-7390/14/20/3819},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Graphs with Large Vertex–Edge Roman {2}-Domination Number
AU  - Ahlam Almulhim
PY  - 2026
DA  - 2026/10/20
JO  - Mathematics
DO  - 10.3390/math14203819
UR  - https://www.mdpi.com/2227-7390/14/20/3819
AB  - For a graph G=(V,E), a function f:V⟶{0,1,2} is a vertex–edge Roman {2}-dominating function (veR2DF) if for every edge xy∈E, either f(x)+f(y)≥1 or ∑z∈N(x)∪N(y)f(z)≥2. The sum ∑v∈Vf(v) is the weight of the function f. The vertex–edge Roman {2}-domination number of G, denoted by γveR2(G), is the minimum weight of a veR2DF of G. In this paper, we prove that for any connected graph G of order n≥4, then γveR2(G)≤n2, and this result is sharp. We also prove Nordhaus–Gaddum inequalities for γveR2(G). In addition, we characterize trees with equal vertex–edge Roman {2}-domination, vertex–edge domination, and domination numbers.
SN  - 2227-7390
ER  -