Restrained Global Defensive Alliances on Some Special Classes of Graphs

Let G = (V (G),E(G)) be a graph. A set S
V is a dominating set if every vertex in V (G)\S is adjacent to at least one vertex in S. A restrained dominating set in G is a set S
V (G) where every vertex in V (G) \ S is adjacent to a vertex in S as well as another vertex in V (G) r S. A defensive alliance in G is a nonempty set of vertices S
V (G) if for every vertex v
S, we have |N[v]
S|
|N(v)
(V (G) \ S)|. A defensive alliance S is called global if it effects every vertex in V (G)\ S, that is, every vertex in V (G)\S is adjacent to at least one member of the alliance S. It is known that graphs may represent dierent situations depending on how certain conditions were used. This study focused on those situations where restrained global defensive alliances were applied. Here, we investigate the formation and properties of restrained global defensive alliances within graphs, specifically focusing on graphs resembling centipede graphs, sunlet graphs, or helm graphs. We analyze how these alliances behave within these graph structures and identify key characteristics, which we label as 'characterizations.' Additionally, we determine the minimum cardinalities of these alliances, referred to as 'restrained global defensive alliance numbers,' which serve the purpose of establishing efficient networks. Through our examination, we aim to provide insights into the dynamics and eciency of restrained global defensive alliances within these graph configurations.