Advances and Applications in Discrete Mathematics

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RESTRAINED GLOBAL OFFENSIVE ALLIANCES IN SOME GRAPHS

Authors

  • Klarice Shaira R. Tan
  • Isagani S. Cabahug, Jr.

Keywords:

dominating set, restrained dominating set, offensive alliance, global offensive alliance, restrained global offensive alliance.

DOI:

https://doi.org/10.17654/0974165825013

Abstract

Let $G=(V(G), E(G))$ be a connected nontrivial graph. A nonempty set $S \subseteq V(G)$ is defined as a restrained global offensive alliance in $G$ if $S$ is a global offensive alliance and the subgraph $G[V(G)-S]$ has no isolated vertices. In this study, we investigate the necessary and sufficient conditions for the existence of a restrained global offensive alliance in various graph structures, including paths, cycles, and specific graph families. Additionally, we explore the minimum cardinality of such alliances within these graph types. Furthermore, this research generates the exact values and characterizations of the restrained global offensive alliance in the join graphs, providing deeper understanding of their properties.

Received: August 19, 2024
Accepted: November 13, 2024

References

S. M. Hedetniemi, S. T. Hedetniemi and P. Kristiansen, Alliances in graphs, J. Combin. Math. Combin. Comput. 48 (2004), 157-177.

G. Chartrand, L. Lesniak and P. Zhang, Graphs and digraphs, Discrete Mathematics and its Application (6th ed.), (2015).

G. S Domke, J. H. Hattingh, S. T. Hedetniemi, R. C. Laskar and L. R. Markus, Restrained domination in graphs, Discrete Mathematics 203(1-3) (1999), 61-69. https://doi.org/10.1016/s0012-365x(99)00016-3

J. M. Sigarreta and J. A. Rodriguez, On the global offensive alliance number of a graph, Discrete Applied Mathematics 157(2) (2009), 219-226.

https://doi.org/10.1016/j.dam.2008.02.007

Soumen Maity and Ajinkya Gaikwad, Offensive Alliances in Graphs, Social Science Research Network, 2023. https://dx.doi.org/10.2139/ssrn.4505390

I. S. Cabahug, Jr. and R. T. Isla, Global offensive alliances in some special classes of graphs, MINDANAWAN Journal of Mathematics (2021), 43-48.

L. F. Consistente and I. S. Cabahug, Jr., 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

L. F. Consistente and I. S. Cabahug, Jr., Restrained global defensive alliances on some special classes of graphs, Asian Research Journal of Mathematics 20(5) (2024), 1-13. https://doi.org/10.9734/arjom/2024/v20i5797

K. S. R. Tan and I. S. Cabahug, Jr., Safe sets in some graph families, Asian Research Journal of Mathematics 18(9) (2022), 1-7.

https://doi.org/10.9734/arjom/2022/v18i930399

D. P. Mangubat and I. S. Cabahug, Jr., On the restrained cost effective sets of some special classes of graphs, Asian Journal of Mathematics 18(8) (2022), 22-34. https://doi.org/10.9734/arjom/2022/v18i830395

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Published

2025-01-02

Issue

Vol. 42 No. 3 (2025)

Section

Articles

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Copyright (c) 2025 PUSHPA PUBLISHING HOUSE, PRAYAGRAJ, INDIA

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How to Cite

RESTRAINED GLOBAL OFFENSIVE ALLIANCES IN SOME GRAPHS. (2025). Advances and Applications in Discrete Mathematics, 42(3), 191-204. https://doi.org/10.17654/0974165825013

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