Chapter 10, Problem 29SE

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.

29.Show that each of these properties is an invariant that iso-morphic simple graphs either both have or both do not have. a) connectedness b)the existence of a Hamilton circuit

c) the existence of an Euler circuit

d) having crossing number C

e) havingnisolated vertices