Geodesic
Signed Total Roman Edge Domination In Graphs
Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 4, pp. 1039-1053.

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Let G = (V,E) be a simple graph with vertex set V and edge set E. A signed total Roman edge dominating function of G is a function f : E →−1, 1, 2 satisfying the conditions that (i) Σ_e^′ ∈ N(e) f(e^′) ≥ 1 for each e ∈ E, where N(e) is the open neighborhood of e, and (ii) every edge e for which f(e) = −1 is adjacent to at least one edge e^′ for which f(e^′) = 2. The weight of a signed total Roman edge dominating function f is ω(f) = Σ_e ∈ E f(e). The signed total Roman edge domination number γ_stR^' (G) of G is the minimum weight of a signed total Roman edge dominating function of G. In this paper, we first prove that for every tree T of order n ≥ 4, γ_stR^' (T) ≥17−2n/5 and we characterize all extreme trees, and then we present some sharp bounds for the signed total Roman edge domination number. We also determine this parameter for some classes of graphs.
Keywords: signed total Roman dominating function, signed total Roman domination number, signed total Roman edge dominating function, signed total Roman edge domination number 
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Asgharsharghi, Leila; Sheikholeslami, Seyed Mahmoud. Signed Total Roman Edge Domination In Graphs. Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 4, pp. 1039-1053. http://geodesic.mathdoc.fr/item/DMGT_2017_37_4_a13/

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