2021/05/11 by Abel Cabrera Martínez, Martinez, Abel Cabrera, Alejandro Estrada‐Moreno +3
Computer Science · #05C69 #05C76 #Advanced Graph Theory Research #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2105.05199
openalex publication_date 2021/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let w=(w0,w1, \…,wl) be a vector of nonnegative integers such that \nw0\≥ 1. Let G be a graph and N(v) the open neighbourhood of v\∈ V(G).\nWe say that a function f: V(G) longrightarrow 0,1,\… ,l is a\nw-dominating function if f(N(v))=\∑u\∈ N(v)f(u)\≥ wi for every\nvertex v with f(v)=i. The weight of f is defined to be\n\ω(f)=\∑v\∈ V(G) f(v). Given a w-dominating function f and any\npair of adjacent vertices v, u\∈ V(G) with f(v)=0 and f(u)>0, the\nfunction fu\→ v is defined by fu\→ v(v)=1,\nfu\→ v(u)=f(u)-1 and fu\→ v(x)=f(x) for every x\∈\nV(G)\∖ u,v . We say that a w-dominating function f is a secure\nw-dominating function if for every v with f(v)=0, there exists u\∈\nN(v) such that f(u)>0 and fu\→ v is a w-dominating function\nas well. The (secure) w-domination number of G, denoted by\n(\γws(G)) \γw(G), is defined as the minimum weight among all\n(secure) w-dominating functions.\n In this paper, we show how the secure (total) domination number and the\n(total) weak Roman domination number of lexicographic product graphs G\∘ H\nare related to \γws(G) or \γw(G). For the case of the secure\ndomination number and the weak Roman domination number, the decision on whether\nw takes specific components will depend on the value of\n\γ(1,0)s(H), while in the case of the total version of these\nparameters, the decision will depend on the value of \γ(1,1)s(H).\n