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Suppose T is a tree. Define T' to be T with all end vertices removed. If x is a vertex of T, then T - x has k components, which I will call T1, ..., Tk. (Therefore, k = 1 if x is an end vertex, k = 0 if x is the only vertex, and otherwise k ≥ 2.) For each i = 1, ..., k, define Di(x) = max dist(x,u) over all vertices u in Ti.
Now here is the problem: prove (or disprove) the following theorem.
Theorem: The following statements about a tree T and a vertex x of T are equivalent.
- x is a center of T.
- x is a center of some longest path in T.
- x is a center of every longest path in T.
- x is a center of T' (in this, assuming n ≥ 3).
- All Di(x) (for i = 1, ..., k) differ by at most 1 (here, assuming deg(x) ≥ 2).
If any part of this theorem is not true, try to find a correct statement.