On the Italian reinforcement number of a digraph

Abstract

The Italian reinforcement number of a digraph is the minimum number of arcs that have to be added to the digraph in order to decrease the Italian domination number. In this paper, we present some new sharp upper bounds on the Italian reinforcement number of a digraph. We also determine the exact values of the Italian reinforcement number of the Cartesian products of directed paths and directed cycles: $ P_2\square P_n $, $ P_3\square P_n $, $ P_3\square C_n $, $ C_3\square P_n $ and $ C_3\square C_n $.