2014/07/12 by Adel P. Kazemi, Kazemi, Adel P.
Mathematics · #05C69 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05C69
paper · pdf · doi:10.48550/arxiv.1407.3363
Accepted in IJMSI for publishing, Jan. 2019
arxiv created 2019/05/01 · arxiv updated 2019/05/02
For any integer k≥ 1 and any graph G=(V,E) with minimum degree at least k-1, we define a function f:V→ \0,1,2\ as a Roman k-tuple dominating function on G if for any vertex v with f(v)=0 there exist at least k and for any vertex v with f(v)≠ 0 at least k-1 vertices in its neighborhood with f(w)=2. The minimum weight of a Roman k-tuple dominating function f on G is called the Roman k-tuple domination number of the graph where the weight of f is f(V)=∑v∈ Vf(v). In this paper, we initiate to study the Roman k-tuple domination number of a graph, by giving some sharp bounds for the Roman k-tuple domination number of a garph, the Mycieleskian of a graph, and the corona graphs. Also finding the Roman k-tuple domination number of some known graphs is our other goal. Some of our results extend these one given by Cockayne and et al. in 2004 for the Roman domination number.