Survey on Roman {2}-Domination
Abstract
:1. Introduction
2. Essential Definitions and Notations
3. Roman -Domination
3.1. General Results
- (i)
- ;
- (ii)
- if and only if ;
- (iii)
- if then ;
- (iv)
- ;
- (v)
- if and only if ;
- (vi)
- if then ;
- (vii)
- if G is connected and then ;
- (viii)
- if then ;
- (ix)
- if then .
3.2. Special Classes of Graphs
- (i)
- Let n be a positive integer greater than 2. Then,
- (ii)
- Let n be a positive integer greater than 3. Then,
- (i)
- If , then
- (ii)
- If , then
- (iii)
- If , , then
- (i)
- , for .
- (ii)
- , for .
- (iii)
- , for and .
- (iv)
- , .
- (i)
- ;
- (ii)
- , and if and only if .
- (i)
- If and for some positive integer r and s, then .
- (ii)
- For an odd integer , .
- (iii)
- For an odd integer , .
- (iv)
- For positive integer , .
3.3. Algorithmic Complexity
4. Roman -Domination Variants
4.1. Perfect Roman -Domination
4.2. Total Roman -Domination
- (i)
- every vertex satisfies that , and
- (ii)
- every vertex is adjacent to at least one vertex .
- (i)
- .
- (ii)
- .
- (i)
- .
- (ii)
- .
- (i)
- For any graph G without isolated vertices, . The bound is sharp and holds for the star graph when .
- (ii)
- For any graph G of order n and ,
- (i)
- For ,
- (ii)
- For , .
- (iii)
- If the graph G is a Hamiltonian graph of order n, then
- (iv)
- If the graph G of order n has a Hamiltonian path, then
4.3. Independent Roman -Domination
4.4. Roman -Reinforcement and Roman -Bondage
4.5. Total Roman -Reinforcement Number
4.6. Restrained Roman -Domination
4.7. Secure Roman -Domination
4.8. Quasi-Total Roman -Domination
- (i)
- .
- (ii)
- .
- (iii)
- If or G is a claw-free graph, then
- (iv)
- For any integer , and .
- (v)
- If , then .
- (vi)
- If , then .
- (vii)
- If G is a graph with no isolated vertex, then
- (a)
- (b)
- if and only if , and
- (c)
- .
- (viii)
- For any graph G with every component of order at least three, then
- (a)
- ,
- (b)
- , and
- (c)
- .
- (ix)
- Gallai-type theorem: For any graph G, .
4.9. Signed Roman -Domination
4.10. Starred Roman -Domination
4.11. The Roman -Domatic Number
- (i)
- .
- (ii)
- if and only if G is isomorphic to the complete graph .
- (iii)
- If , then , and the equality holds if and only if or G is the complete graph.
- (i)
- There is one 4-regular graph of order for which .
- (ii)
- There is no 7-regular graph of order for which .
4.12. The Covering Roman -Domination
4.13. The Global Roman -Domination
4.14. Outer-Independent Roman -Domination
4.15. -Domination
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Almulhim, A.; Al Subaiei, B.; Mondal, S.R. Survey on Roman {2}-Domination. Mathematics 2024, 12, 2771. https://doi.org/10.3390/math12172771
Almulhim A, Al Subaiei B, Mondal SR. Survey on Roman {2}-Domination. Mathematics. 2024; 12(17):2771. https://doi.org/10.3390/math12172771
Chicago/Turabian StyleAlmulhim, Ahlam, Bana Al Subaiei, and Saiful Rahman Mondal. 2024. "Survey on Roman {2}-Domination" Mathematics 12, no. 17: 2771. https://doi.org/10.3390/math12172771