The Number of Minimum Roman and Minimum Total Dominating Sets for Some Chessboard Graphs

Abstract

In this paper, both the roman domination number and the number of minimum roman dominating sets are found for any rectangular rook’s graph. In a similar fashion, the roman domination number and the number of minimum roman dominating sets are found on the square bishop’s graph for odd board sizes. Also found are the number of minimum total dominating sets associated with the light-colored squares when n ≡ 1(mod12)  (with n>1), and same for the dark-colored squares when n ≡ 7(mod12) .