Abstract
Let G=(V,E) be a graph. A function f:V→\{0,1,2\}, if ∀u for which f(u)=0 is adjacent to ∃v for which f(v)=2, is called a Roman dominating function, and called in short terms RDF. The weight of an RDF f is f(V)=∑_(v∈V)▒f(v) . The Roman domination number of a graph G, denoted by γ_R (G), is the minimum weight of an RDF on G. This paper presents the results for Roman domination numbers of the Mycielski graphs obtained through Mycielski's construction of the comet, double comet, and comb graphs. An algorithm to determine the Roman domination number of any given graph is also provided.
Keywords
References
- Chambers, E. W., Kinnersley, B., Prince, N., & West, D. B. (2009). Extremal problems for Roman domination. SIAM Journal on Discrete Mathematics, 23(3), 1575-1586. https://doi.org/10.1137/070699688
- Cockayne, E. J., Dreyer Jr, P. A., Hedetniemi, S. M., & Hedetniemi, S. T. (2004). Roman domination in graphs. Discrete Mathematics, 278(1-3), 11-22. https://doi.org/10.1016/j.disc.2003.06.004
- Dreyer, P. A. (2000). Applications and variations of domination in graphs. (PhD), Rutgers University, New Jersey.
- Durgun, D. D., & Toprakkaya, E. N. (2021). Roman domination of the Comet, Double Comet, and Comb Graphs. arXiv preprint arXiv:2102.07902. https://doi.org/10.48550/arXiv.2102.07902
- Durgun, D. D., & Toprakkaya, E. N. (2023). Roman domination on some graphs. Journal of Modern Technology and Engineering, 8(2), 96-104.
- Haynes, T. W., Hedetniemi, S., & Slater, P. (1998). Fundamentals of domination in graphs. CRC Press. https://doi.org/10.1201/9781482246582
- Henning, M. A., & Hedetniemi, S. T. (2003). Defending the Roman Empire - A new strategy. Discrete Mathematics, 266(1-3), 239-251. https://doi.org/10.1016/S0012-365X(02)00811-7
- Kazemi, A. P. (2012). Roman domination and Mycielski’s structure in graphs. Ars Combinatoria, 106, 277-287.
- Liedloff, M., Kloks, T., Liu, J., & Peng, S. (2008). Efficient algorithms for Roman domination on some classes of graphs. Discrete Applied Mathematics, 156(18), 3400-3415. https://doi.org/10.1016/j.dam.2008.01.011
- Stewart, I. (1999). Defend the Roman Empire!, Scientific American, 281(6), 136-138. https://doi.org/10.1038/scientificamerican1299-136
- West, D. B. (2001). Introduction to graph theory. Prentice Hall Upper Saddle River.
Abstract
G=(V,E) bir graf olsun. f(u)=0 olan her u tepesinin, f(v)=2 olan en az bir v tepesine bitişik olması koşulunu karşılayan bir Roma baskınlık fonksiyonu (RDF) f:V→\{0,1,2\}. Bir RDF f’in ağırlığı f(V)=∑_(v∈V)▒f(v) . Bir G grafının Roma baskınlık sayısı, γ_R (G) ile gösterilir, G’de bir RDF’nin minimum ağırlığıdır. Bu çalışmada, Mycielski'nin kuyruklu yıldız, çift kuyruklu yıldız ve tarak graflarını oluşturmasıyla elde edilen Mycielski graflarının Roma baskınlık sayılarına ilişkin sonuçlarını sunmaktadır. Ayrıca, herhangi bir grafın Roma baskınlık sayısını belirleyen bir algoritma sağlanmıştır.
Keywords
References
- Chambers, E. W., Kinnersley, B., Prince, N., & West, D. B. (2009). Extremal problems for Roman domination. SIAM Journal on Discrete Mathematics, 23(3), 1575-1586. https://doi.org/10.1137/070699688
- Cockayne, E. J., Dreyer Jr, P. A., Hedetniemi, S. M., & Hedetniemi, S. T. (2004). Roman domination in graphs. Discrete Mathematics, 278(1-3), 11-22. https://doi.org/10.1016/j.disc.2003.06.004
- Dreyer, P. A. (2000). Applications and variations of domination in graphs. (PhD), Rutgers University, New Jersey.
- Durgun, D. D., & Toprakkaya, E. N. (2021). Roman domination of the Comet, Double Comet, and Comb Graphs. arXiv preprint arXiv:2102.07902. https://doi.org/10.48550/arXiv.2102.07902
- Durgun, D. D., & Toprakkaya, E. N. (2023). Roman domination on some graphs. Journal of Modern Technology and Engineering, 8(2), 96-104.
- Haynes, T. W., Hedetniemi, S., & Slater, P. (1998). Fundamentals of domination in graphs. CRC Press. https://doi.org/10.1201/9781482246582
- Henning, M. A., & Hedetniemi, S. T. (2003). Defending the Roman Empire - A new strategy. Discrete Mathematics, 266(1-3), 239-251. https://doi.org/10.1016/S0012-365X(02)00811-7
- Kazemi, A. P. (2012). Roman domination and Mycielski’s structure in graphs. Ars Combinatoria, 106, 277-287.
- Liedloff, M., Kloks, T., Liu, J., & Peng, S. (2008). Efficient algorithms for Roman domination on some classes of graphs. Discrete Applied Mathematics, 156(18), 3400-3415. https://doi.org/10.1016/j.dam.2008.01.011
- Stewart, I. (1999). Defend the Roman Empire!, Scientific American, 281(6), 136-138. https://doi.org/10.1038/scientificamerican1299-136
- West, D. B. (2001). Introduction to graph theory. Prentice Hall Upper Saddle River.
Details
| Primary Language | English |
|---|---|
| Subjects | Applied Mathematics (Other) |
| Journal Section | Natural Sciences and Mathematics / Fen Bilimleri ve Matematik |
| Authors | |
| Publication Date | August 31, 2024 |
| Submission Date | February 25, 2024 |
| Acceptance Date | May 30, 2024 |
| Published in Issue | Year 2024 |