Secure w-Domination in Graphs

This paper introduces a general approach to the idea of protection of graphs, which encompasses the known variants of secure domination and introduces new ones. Specifically, we introduce the study of secure w-domination in graphs, where w=(w0,w1,…,wl) is a vector of nonnegative integers such that w0≥1. The secure w-domination number is defined as follows. Let G be a graph and N(v) the open neighborhood of v∈V(G). We say that a function f:V(G)⟶{0,1,…,l} is a w-dominating function if f(N(v))=∑u∈N(v)f(u)≥wi for every vertex v with f(v)=i. The weight of f is defined to be ω(f)=∑v∈V(G)f(v). Given a w-dominating function f and any pair of adjacent vertices v,u∈V(G) with f(v)=0 and f(u)>0, the function fu→v is defined by fu→v(v)=1, fu→v(u)=f(u)−1 and fu→v(x)=f(x) for every x∈V(G)\{u,v}. We say that a w-dominating function f is a secure w-dominating function if for every v with f(v)=0, there exists u∈N(v) such that f(u)>0 and fu→v is a w-dominating function as well. The secure w-domination number of G, denoted by γws(G), is the minimum weight among all secure w-dominating functions. This paper provides fundamental results on γws(G) and raises the challenge of conducting a detailed study of the topic.