# Generalized geography game graph

I'm studying the Sipser textbook for my theory of complexity class. In a part of the book (i.e., Space Complexity chapter), for showing that Generalized Geography game is PSPACE-complete, the author has given an argument to model this game with TQBF problem (which is proven to be PSPACE-complete).

In a step of the argument, he has tried to construct a directed graph by using the definitions of TQBF problem (universal and existential quantifiers). What he has come up with is the following graph:

1. I, actually, don't get it why the diamond structure has been used for showing choice possibilities of player $$x_i$$. Couldn't we use a simple structure like what we do on binary trees?
1. How would it be if we use a hexagon shape instead of diamond for showing choices of $$x_i$$?
Your graph based on a binary tree would have exponential size (it would require at least $$2^k$$ nodes), which is probably no good for the reduction; an exponential blow-up in the size of the problem instance isn't ok.