Criticality indices of Roman domination of paths and cycles

For a graph G = (V,E), a Roman dominating function on G is a function f : V (G) → {0, 1, 2} such that every vertex u for which f(u) = 0 is adjacent to at least one vertex v with f(v) = 2. The weight of a Roman dominating function is the value f(V (G)) = ∑ u∈V (G) f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G, denoted by γR (G). The removal criticality index of a graph G is defined as ciR(G) = ( ∑ e∈E(G)(γR (G)− γR (G− e))/ |E(G)| and the adding criticality index of G is defined as ci+R(G) = ( ∑ e∈E(G)(γR (G)−γR (G + e))/ ∣∣E(G)∣∣ where G is the complement graph of G. In this paper, we determine the criticality indices of paths and cycles.