Fair Secure Roman Dominating Function in Graphs

Let G=(V(G),E(G)) be a graph and let ϕ:V(G)→{0,1,2} be a function on G. For each i∈{0,1,2}, let V_i={v∈V(G):ϕ(v)=i}. Then ϕ can be represented as ϕ=(V_0,V_1,V_2). A function ϕ is a fair secure Roman dominating function (FScRDF) on G provided that for every v∈V_0, there exists u∈V_2 such that d_G (u,v)=1, ϕ^*=(V_0∖{v}, V_1∪{v,u}, V_2∖{u}) is a Roman dominating function (RDF) on G, and for every x,y∈V_0, ∣N_G (x)∩V_2∣=∣N_G (y)∩V_2∣≥1. The weight of FScRDF ϕ on G, denoted by ω_G^FScR (ϕ), is defined as the sum ω_G^FScR (ϕ)=∑_(x∈V(G))〖ϕ(x)=∣〗 V_1∣+2∣V_2∣. The fair secure Roman domination number of G is defined as the minimum weight of an FScRDF ϕ on G, and is denoted by γ_FScR (G), that is, γ_FScR (G)=min⁡{ω_G^FScR (ϕ):ϕ" is an FScRDF on " G}. Every FScRDF ϕ on G that satisfies ω_G^FScR (ϕ)=γ_FScR (G) is called a γ_FScR-function on G. In this paper, the authors introduce the idea of fair secure Roman domination in graphs as a new parameter and discuss some important combinatorial results.