9.7. Positive integers from 1 to 100 are arranged in a circle in such a way that each number is either greater than both of its neighboring numbers or less than both of its neighboring numbers. A pair of adjacent numbers is called "good" if, after removing them, the above property still holds. How many "good" pairs of neighbors can there be at a minimum?
9.7. The minimum possible number of "good" neighboring pairs is 51.

For example, first arrange the numbers from 1 to 100 in a clockwise direction in increasing order on a circle, then swap the positions of 2 and 3, 4 and 5, ..., 98 and 99. In the resulting arrangement \(1, 3, 2, 5, 4, \cdots, 99, 98, 100\), there are exactly 51 pairs of "good" neighbors:
\[
\begin{array}{l}
(1,3), (3,2), (5,4), (7,6), \cdots, (97,96), \\
(99,98), (98,100).
\end{array}
\]

Next, we prove that the number of "good" neighboring pairs is at least 51. To do this, we first prove that in any two intersecting "pairs," at least one pair is good.

Assume there are 5 consecutive numbers \(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\) such that \((a_{2}, a_{3})\) and \((a_{3}, a_{4})\) are not good.

Without loss of generality, we can assume \(a_{1} > a_{2} < a_{3} > a_{4} < a_{5}\). Since \((a_{2}, a_{3})\) is not good, we have \(a_{1} < a_{2} < a_{3} > a_{4} > a_{5}\). Similarly, since \((a_{3}, a_{4})\) is not good, we have \(a_{1} > a_{2} > a_{3} < a_{4} < a_{5}\). This leads to a contradiction: \(a_{1} < a_{2} < a_{3} > a_{4} > a_{5}\) and \(a_{1} > a_{2} > a_{3} < a_{4} < a_{5}\) cannot both be true. Therefore, at least one of the pairs \((a_{2}, a_{3})\) or \((a_{3}, a_{4})\) must be good.

This means there are at least 50 good neighboring pairs. If there are exactly 50 good pairs, then "good" and "not good" neighboring pairs must alternate. However, if we consider the numbers \(a_{i-2}, a_{i-1}, 100, a_{i}, a_{i+1}\) around the number 100, it is easy to see that the pair \((a_{i}, a_{i+1})\) must be good because \(100 > a_{i} > a_{i+1}\) or \(100 < a_{i} < a_{i+1}\). Similarly, the pair \((a_{i-2}, a_{i-1})\) must also be good. This shows that "good" and "not good" neighboring pairs cannot alternate.

Thus, the number of "good" neighboring pairs is at least 51.