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

Paul A Burchett

doi:10.4236/ojdm.2020.101004

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) .

Highlights

The m × n rook’s graph, denoted by Rm,n, is formed by associating the squares of our m × n board with vertices
Found are the number of minimum total dominating sets associated with the light-colored squares when n ≡ 1(mod12), and same for the dark-colored squares when n ≡ 7
This paper’s findings imply that for n ≡ 1(mod12), the total domination number of the subgraph associated with the light-colored squares is 2(n −1), and the same when n ≡ 7 for the subgraph associated with the dark-colored squares

Summary

Introduction

The m × n rook’s graph, denoted by Rm,n , is formed by associating the squares of our m × n board with vertices. Γ r ( Bn ) denotes the roman domination number on the square, n × n bishop’s graph. On the bishop’s graph, a set can be said to be a total dominating set if and only if every square is attacked. This paper’s findings imply that for n ≡ 1(mod12) (with n > 1 ), the total domination number of the subgraph associated with the light-colored squares is 2(n −1) , and the same when n ≡ 7 (mod12) for the subgraph associated with the dark-colored squares. For more on the total domination number, a good book on the current literature is [11]

Rook’s Graph Results for Roman Domination

The Roman Domination Number on the Bishop’s Graph and the Count for Odd n

Total Domination Section

Conclusion