Properties of double Roman domination on cardinal products of graphs

Double Roman domination is a stronger version of Roman domination that doubles the protection. The areas now have 0 , 1 , 2 or 3 legions. Every attacked area needs 2 legions for its defence, either their own, or borrowed from 1 or 2 neighbouring areas, which still have to keep at least 1 legion to themselves. The minimal number of legions in all areas together is equal to the double Roman domination number. In this paper we determine an upper bound and a lower bound for double Roman domination numbers on cardinal product of any two graphs. Also we determine the exact values of double Roman domination numbers on P 2  ×  G (for many types of graph G ). Also, the double Roman domination number is found for P 2  ×  P n , P 3  ×  P n , P 4  ×  P n , while upper and lower bounds are given for P 5  ×  P n and P 6  ×  P n . Finally, we will give a case study to determine the efficiency of double protection. We will compare double Roman domination versus Roman domination by running a simulation of a battle.