Note: On the 2-rainbow domination in graphs
Discrete Applied Mathematics
Discrete Applied Mathematics
Handbook of Product Graphs, Second Edition
Handbook of Product Graphs, Second Edition
On the Roman domination in the lexicographic product of graphs
Discrete Applied Mathematics
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A k-rainbow dominating function of a graph G is a map f from V(G) to the set of all subsets of {1,2,...,k} such that {1,...,k}=@?"u"@?"N"("v")f(u) whenever v is a vertex with f(v)=0@?. The k-rainbow domination number of G is the invariant @c"r"k(G), which is the minimum sum (over all the vertices of G) of the cardinalities of the subsets assigned by a k-rainbow dominating function. We focus on the 2-rainbow domination number of the lexicographic product of graphs and prove sharp lower and upper bounds for this number. In fact, we prove the exact value of @c"r"2(G@?H) in terms of domination invariants of G except for the case when @c"r"2(H)=3 and there exists a minimum 2-rainbow dominating function of H such that there is a vertex in H with the label {1,2}.