Iranian Journal of

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Pushpam P R L, Srilakshmi N. Global Weak Roman Domination in Graphs. IJMSI 2026; 21 (1) :217-228
URL: http://ijmsi.ir/article-1-2163-en.html

A Roman dominating function (RDF) of a graph G = (V, E) is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. A vertex u with f (u) = 0 is said to be undefended if it is not adjacent to a vertex with f(v) > 0. For a graph G, a function f : V (G)→ {0, 1, 2} is said to be a weak Roman dominating function (WRDF) if each vertex u with f (u) = 0 is adjacent to a vertex v with f(v) > 0 such that the function f′: V (G) → {0, 1, 2} defined by f′(u) = 1, f′(v) = f (v)- 1 and f′(w) = f (w) if w ∈ V - {u, v}, has no undefended vertex. The weight of f is defined to be the value f(V ) = ∑_u∈V f (u). The minimum weight of a weak Roman dominating function of a graph G is called the weak Roman domination number of G and is denoted by γ_r(G). A WRDF with weight γ_r(G) is called a γ_r(G)-function. A set S ⊆V is a global dominating set if S dominates both G and its complement G. The global domination number γ_g(G) of a graph G is the minimum cardinality of a global dominating set S. We extend the idea of global domination to weak Roman domination as follows: For a graph G, the function f : V (G) → {0, 1, 2} is a global weak Roman dominating function (GWRDF) if f is a WRDF for both G and its complement G. The weight of a global weak Roman dominating function is the value
f (V ) = ∑_u∈V f (u). The minimum weight of a global weak Roman dominating function of a graph G is called the global weak Roman domination number of G and is denoted by γ_gr(G). In this paper, we initiate a study of this parameter.