Tweedle Graph

Tweedledee and Tweedledum are in a fierce battle playing a complicated graph game. The game is played on an undirected bidirectional simple graph of $N$ vertices (enumerated $1$ trhough $N$) and $(N \cdot (N-1))/2$ edges. We'll denote the beauty of vertex $u$ by $B_u$. The game goes as follows: - First Twedledee places a token in one of the vertices of the graph. - Then players alternate