Geodesic
Total Roman {2}-Dominating Functions in Graphs
Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 3, pp. 937-958.

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A Roman 2-dominating function (R2F) is a function f : V → 0, 1, 2 with the property that for every vertex v ∈ V with f(v) = 0 there is a neighbor u of v with f(u) = 2, or there are two neighbors x, y of v with f(x) = f(y) = 1. A total Roman 2-dominating function (TR2DF) is an R2F f such that the set of vertices with f(v) gt; 0 induce a subgraph with no isolated vertices. The weight of a TR2DF is the sum of its function values over all vertices, and the minimum weight of a TR2DF of G is the total Roman 2-domination number γtR2(G). In this paper, we initiate the study of total Roman 2-dominating functions, where properties are established. Moreover, we present various bounds on the total Roman 2-domination number. We also show that the decision problem associated with γtR2(G) is possible to compute this parameter in linear time for bounded clique-width graphs (including trees).
Keywords: Roman domination, Roman {2}-domination, total Roman {2}-domination 
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Ahangar, H. Abdollahzadeh; Chellali, M.; Sheikholeslami, S.M.; Valenzuela-Tripodoro, J.C. Total Roman {2}-Dominating Functions in Graphs. Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 3, pp. 937-958. http://geodesic.mathdoc.fr/item/DMGT_2022_42_3_a15/

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