Geodesic
Signed Roman Edge k -Domination in Graphs
Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 1, pp. 39-53

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Let k ≥ 1 be an integer, and G = (V, E) be a finite and simple graph. The closed neighborhood N_G [e] of an edge e in a graph G is the set consisting of e and all edges having a common end-vertex with e. A signed Roman edge k-dominating function (SREkDF) on a graph G is a function f : E →−1, 1, 2 satisfying the conditions that (i) for every edge e of G, Σ_ x ∈ N_G [e] f(x) ≥ k and (ii) every edge e for which f(e) = −1 is adjacent to at least one edge e^′ for which f(e^′) = 2. The minimum of the values Σ_e ∈ E f(e), taken over all signed Roman edge k-dominating functions f of G is called the signed Roman edge k-domination number of G, and is denoted by γ_sRk^' (G). In this paper we initiate the study of the signed Roman edge k-domination in graphs and present some (sharp) bounds for this parameter.
Keywords: signed Roman edge k -dominating function, signed Roman edge k -domination number 
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     title = {Signed {Roman} {Edge} k {-Domination} in {Graphs}},
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Asgharsharghi, Leila; Sheikholeslami, Seyed Mahmoud; Volkmann, Lutz. Signed Roman Edge k -Domination in Graphs. Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 1, pp. 39-53. http://geodesic.mathdoc.fr/item/DMGT_2017_37_1_a3/