Secure Italian domination in graphs

Abstract

An Italian dominating function (IDF) on a graph G is a function f:V(G)rightarrow {0,1,2} such that for every vertex v with f(v)=0, the total weight of f assigned to the neighbours of v is at least two, i.e., sum _{uin N_G(v)}f(u)ge 2. For any function f:V(G)rightarrow {0,1,2} and any pair of adjacent vertices with f(v) = 0 and u with f(u) > 0, the function f_{urightarrow v} is defined by f_{urightarrow v}(v)=1, f_{urightarrow v}(u)=f(u)-1 and f_{urightarrow v}(x)=f(x) whenever xin V(G){setminus }{u,v}. A secure Italian dominating function on a graph G is defined as an IDF f which satisfies that for every vertex v with f(v)=0, there exists a neighbour u with f(u)>0 such that f_{urightarrow v} is an IDF. The weight of f is omega (f)=sum _{vin V(G) }f(v). The minimum weight among all secure Italian dominating functions on G is the secure Italian domination number of G. This paper is devoted to initiating the study of the secure Italian domination number of a graph. In particular, we prove that the problem of finding this parameter is NP-hard and we obtain general bounds on it. Moreover, for certain classes of graphs, we obtain closed formulas for this novel parameter.