A note on outer-connected hop Roman dominating function in graphs - Casinillo, Leomarich F (Diponegoro University, 2025)
Let G=(V(G), E(G)) be a simple, connected, and finite graph with vertex set V(G) and edge set E(G). Let φ: V(G) → {0, 1, 2} be an HRDF on G, and for each i{0, 1, 2}, let V_i={uV(G): φ(u)=i}. A function φ=(V_0, V_1, V_2) is an outer-connected hop Roman dominating function (OcHRDF) on G if, for every vV_0, there exists uV_2 such that d_G(u, v)=2 and either V_1=V(G) or the sub-graph V_0 is connected. The weigth of OcHRDF φ denoted by ω_G^chR(φ) and defined by ω_G^chR(φ)=_vV(G)φ(v)=|V_1|+2|V_2|. The outer-connected hop Roman domination number of G is denoted by γ_chR(G) and defined by γ_chR(G)=min{ω_G^chR(φ): φ is an OcHRDF on G}. Moreover, any OcHRDF φ on (G) with γ_chR(G)=ω_G^chR(φ) is called γ_chR-function on G. In this paper, a new restricted parameter of a hop Roman domination in graphs is introduced, and some combinatorial properties are discussed.