Geodesic
Weak roman domination in graphs
Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 1, pp. 161-170

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Let G = (V,E) be a graph and f be a function f:V → 0,1,2. A vertex u with f(u) = 0 is said to be undefended with respect to f, if it is not adjacent to a vertex with positive weight. The function f is 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 → 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 w(f) = ∑_v ∈ Vf(v). The weak Roman domination number, denoted by γ_r(G), is the minimum weight of a WRDF in G. In this paper, we characterize the class of trees and split graphs for which γ_r(G) = γ(G) and find γ_r-value for a caterpillar, a 2×n grid graph and a complete binary tree.
Keywords: domination number, weak Roman domination number 
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Roushini Leely Pushpam, P.; Malini Mai, T. Weak roman domination in graphs. Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 1, pp. 161-170. http://geodesic.mathdoc.fr/item/DMGT_2011_31_1_a9/