Roman domination number on cardinal product of paths and cycles

In this paper, the authors have determined certain upper and lower bounds for Roman domination numbers on cardinal products for any two graphs and some exact values for the cardinal product of paths and cycles. Roman domination was named after the historical fact that Roman legions were distributed across regions during the reign of the Roman Emperor Constantine in the 4th century A.D. Some areas had 1 or 2 legions, some had no legions, but every area had at least 1 neighboring area with 2 legions. Roman domination is used even today, not only in military situations, but also in protecting some locations against fire or crime