For a simple undirected connected graph G = ( V, E ), a maximal Roman dominating function (MRDF) of G is a function f : V ( G ) → {0, 1, 2} with the following properties: ( i ) For every vertex v ∈ { v ∈ V | f ( v ) = 0}, there exists a vertex u ∈ N ( v ) such that f ( u ) = 2. ( ii ) The set { v ∈ V|f ( v ) = 0} is not a dominating set of G ; In other words, there exists a vertex v ∈ { v ∈ V | f ( v ) ≠ 0} such that N ( v ) ∩ { u ∈ V | f ( u ) = 0} ∅ . The weight of an MRDF of G is the sum of its function values …