Diameter of Graphs: Why 2 & Exceptions

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The diameter of the complement of a tree is typically 2 because the maximum distance between any two vertices in the complement is minimized by the tree's structure. In a tree, any two vertices are connected by a unique path, and adding edges in the complement connects distant vertices directly. Exceptions occur in specific configurations, such as when the tree is a star or has a high degree of connectivity, which can lead to a larger diameter. Understanding these exceptions requires analyzing the specific structure of the tree and its complement. Overall, the general rule holds true for most tree complements, with notable exceptions based on their topology.
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Explain why the diameter of almost every complement of a tree will be 2 and find all exceptions to this rule
 
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