The solution for the N queens problem was achieved using a backtracking algorithm that explores all possible configurations of queens on the chessboard and eliminates invalid configurations until a valid solution is found.
In this solution, the values used for the edges and vertices of the graph are based on the number of queens in the N queens problem. The vertices of the graph represent the columns of the chessboard, and are numbered from 0 to N-1. Each vertex is connected to every other vertex by an edge, forming a complete graph.
To reduce the number of potential solutions, we use the constraint that there can only be one queen in each row. We start by placing a queen in the first row, and then use the graph to explore all possible placements of the remaining queens in the remaining rows. For each placement, we check if it is valid by ensuring that no two queens are in the same column or on the same diagonal. If a placement is valid, we place the queens on the board and return true, indicating that a solution has been found. If no solution is found, we return false.
The edges of the graph are used to determine the possible placements of the next queen based on the positions of the queens that have already been placed. For each queen that has been placed, we add an edge between its column and every other column that is not in the same row or diagonal. This forms a subgraph of the complete graph that represents the valid placements for the next queen. We then explore each of these valid placements recursively until we find a solution or exhaust all possibilities.