Roman {2}-domination

In this paper, we initiate the study of a variant of Roman dominating functions. For a graph G = (V, E), a Roman {2}-dominating function f : V -> {0, 1, 2} has the property that for every vertex v is an element of V with f (v) = 0, either v is adjacent to a vertex assigned 2 under f, or v is adjacent to least two vertices assigned 1 under f. The weight of a Roman {2}-dominating function is the sum Sigma(v is an element of V)f (v), and the minimum weight of a Roman {2}-dominating function f is the Roman {2}-domination number. First, we present bounds relating the Roman {2}-domination number to some other domination parameters. In particular, we show that the Roman {2}-domination number is bounded above by the 2-rainbow domination number. Moreover, we prove that equality between these two parameters holds for trees and cactus graphs with no even cycles. Finally, we show that associated decision problem for Roman {2}-domination is NP-complete, even for bipartite graphs. (C) 2015 Elsevier B.V. All rights reserved.