Publications Repository - Gdańsk University of Technology

An Italian dominating function (IDF) on a graph G is a function f:V(G)→{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., ∑u∈NG(v)f(u)≥2. For any function f:V(G)→{0,1,2} and any pair of adjacent vertices with f(v)=0 and u with 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) whenever x∈V(G)∖{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 fu→v is an IDF. The weight of f is ω(f)=∑v∈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.

Authors

• dr inż. Magda Dettlaff link open in new tab ,
• dr inż. Magdalena Lemańska link open in new tab ,
• Juan A. RODRíGUEZ-VELáZQUEZ