On Roman balanced domination of graphs - Mingyu Zhang; Junxia Zhang (AIMS Mathematics)
Let $ G $ be a graph with vertex set $ V $. A function $ f $ : $ V→ \{-1, 0, 2\} $ is called a Roman balanced dominating function (RBDF) of $ G $ if $ \sum_{u\in N_G[v]}f(u) = 0 $ for each vertex $ v\in V $. The maximum (resp. minimum) Roman balanced domination number $ γ^{M}_{Rb}(G) $ (resp. $ γ^{m}_{Rb}(G) $) is the maximum (resp. minimum) value of $ \sum_{v\in V} f(v) $ among all Roman balanced dominating functions $ f $. A graph $ G $ is called $ Rd $-balanced if $ γ^{M}_{Rb}(G) = γ^{m}_{Rb}(G) = 0 $. In this paper, we obtain several upper and lower bounds on $ γ^{M}_{Rb}(G) $ and $ γ^{m}_{Rb}(G) $ and further determine several classes of $ Rd $-balanced graphs.