# People and acquaintances - simple graph theoretical question

I have taken the following problem from an introductory text on graph theory. Somehow I am missing something about this problem, as the statement seems wrong to me.

There is a group of $n$ people $A_1, \ldots,A_n$ in which some people know each other and any two people who do not know each other would have common acquaintance. Suppose that $A_1$ and $A_2$ know each other but do not have common acquaintance. Show that the acquaintances of $A_1$ are as many as those of $A_2.$

It is obvious how to model this question in the graph theoretical framework. What I am wondering is why isn't $P_3$ (picking a leaf and a 2-vertex for $A_1$ and $A_2$) a counterexample. The two leaf nodes do not know each other but have common friend, but $A_1$ and $A_2$ do not have the same number of acquaintances.

What am I missing?

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Google told me that this question - with a solution - can be found in the book Graph Theory by Bin Xiong, Zhongyi Zheng on page 5. However, I still cannot find flaw in your counterexample, even after comparing it with their alleged proof. Perhaps they have forgotten to add some assumption? –  Martin Sleziak Oct 8 '11 at 9:20
You have found a counter-example, so what you have written here is wrong. For more guessing on the authors' intention, we would need more context. –  Phira Oct 8 '11 at 9:21