For a nontrivial connected graph  a nonempty subset  is a total offensive alliance in G if T is an offensive alliance and every vertex in T has at least one neighbor within T. The minimum cardinality of a total offensive alliance in G is called the total offensive alliance number of G, denoted by  In this paper, we investigate the total offensive alliance of total graphs generated from graphs with maximum degree 2 and present the characterization of total offensive alliance and their corresponding total offensive alliance number.

total graph, offensive alliance, total offensive alliance, total offensive alliance number.

Received: February 7, 2025; Accepted: March 3, 2025; Published: March 12, 2025

How to cite this article: Maxene S. Hablo and Isagani S. Cabahug Jr., Total offensive alliance of total graphs generated from graphs with maximum degree 2, Advances and Applications in Discrete Mathematics 42(4) (2025), 391-399. https://doi.org/10.17654/0974165825025

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:
[1] M. Behzad, A characterization of total graphs, American Mathematics Society (1970), 383-389. https://doi.org/10.2307/2037344.
[2] I. S. Cabahug Jr. and L. F. Consistente, Restrained global defensive alliances in graphs, European Journal of Pure and Applied Mathematics 17(3) (2024), 2196-2209. https://doi.org/10.29020/nybg.ejpam.v17i3.5156.
[3] I. S. Cabahug Jr. and L. F. Consistente, Restrained global defensive alliances on some special classes of graphs, Asian Research Journal of Mathematics 20(50) (2024), 1-13. https://doi.org/10.9734/arjom/2024/v20i5797.
[4] I. S. Cabahug Jr. And R. T. Isla, Global offensive alliances in some special classes of graphs, The Mindanawan Journal of Mathematics 2(1) (2011), 43-48.
https://journals.msuiit.edu.ph/tmjm/article/view/17.
[5] G. Chartrand, L. Lesniak and P. Zhang, Graphs and digraphs, Chapman and Hall/CRC Press, 2015. https://doi.org/10.1201/b14892.
[6] O. Favaron, G. Fricke, W. Goddard, S. M. Hedetniemi, S. T. Hedetniemi, P. Kristiansen, R. C. Laskar and D. Skaggs, Offensive alliances in graphs, Graph 24 (2004), 263-275. http://eudml.org/docc/270644.
[7] A. Gaikwad and S. Maity, Offensive alliances in graphs, 2022.
https://doi.org/10.48550/arXiv.2208.02992.
[8] M. S. Hablo and I. S. Cabahug Jr., Total offensive alliances on some graphs, Asian Research Journal of Mathematics 20(9) (2024), 120-31.
https://doi.org/10.9734/arjom/2024/v20i9835.
[9] S. M. Hedetniemi and P. Kristiansen, Alliances in graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 48 (2004), 57-177.
[10] K. Ouazine, H. Slimani and A. K. Tari, Alliances in graphs: Parameters, properties and applications- a survey. https://doi.ord/10.1016/j.akcej.2017.05.002.