Strong defensive alliances on graph operators

¹ Facultad de Ciencias y Tecnologías de la Información, Universidad Autónoma de Guerrero, Las Colinas 37A, Col. Fracc. Las playas, Acapulco, Guerrero, México
² Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame 5, Col. La Garita, Acapulco, Guerrero, México

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 17 April 2025
Accepted: 10 February 2026

Abstract

If G = (V(G), E(G)) is a simple connected graph with vertex set V(G) and edge set E(G), we say that a subset D ⊆ V(G) is a strong defensive alliance if for every vertex v ∈ D the condition δ_D(v) ≥ δ_D̅(v) holds. The strong defensive alliance number α(G) is defined as the minimum cardinality among all the strong defensive alliances. A unitary operator of graphs 𝒪 assigns to each graph G a graph 𝒪(G). A few examples of unitary operators of graphs are: Subdivision S(G), R(G), Middle Q(G), Total T(G), and Central 𝒞(G). In this paper we determine the exact values of α(S(G)) and α(R(G)). We also characterize the graphs G for which the number of strong defensive alliances is 1, 2, or 3 in Q(G) and T(G). We also we give tight bounds for α(S̅(G̅)), α(Q(G)), α(Q̅(G̅)), α(T(G)), and α(𝒞(G)).