On the total version of the covering Italian domination problem

Abstract

Given a graph G without isolated vertices, a function f : V(G) → {0, 1, 2} is a covering total Italian dominating function if (i) the set of vertices labeled with 0 forms an independent set; (ii) every vertex labeled with 0 is adjacent to two vertices labeled with 1 or to one vertex labeled with 2; and (iii) the set of vertices labeled with 1 or 2 forms a total dominating set. The covering total Italian domination number of G is the smallest possible value of the sum ∑ v∈V(G) f (v) among all possible covering total Italian dominating functions f on V(G). The concepts above are introduced in this article, and the study of its combinatorial and computational properties is initiated. Specifically, we show several relationships between such parameter and other domination related parameters in graphs. We also prove the NP-completeness of the related decision problem for bipartite graphs, and present some approximation results on computing our parameter. In addition, we compute the exact value of the covering total Italian domination number of some graphs with emphasis on some Cartesian products.