Chapter 10, Problem 27SE

A simple graph can be used to determine the minimum number of queens on a chessboard that control the entire chessboard. An $n\times n$chessboard hasn²squares in an $n\times n$configuration. A queen in a given position controls all squares in the same row, the same column, and on the two diagonals containing this square, as illustrated. The appropriate simple graph hasn²vertices, one for each square, and two vertices are adjacent if a queen in the square represented by one of the vertices controls the square represented by the other vertex.

& 27. Find the minimum number of queens controllingan $n**\#x00D7;n$chessboard fora)n= 3.

b) n= 4.

c)n= 5.