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๐ฐ Case 1: The three close pairs share a common vertex รขยย e.g., a star of 3 edges: \( AB, AC, AD \). But we only have 3 edges รขยย this uses all 3 close pairs incident to A. Then consider trio \( B,C,D \): no edges among them (unless present), so no close pairs รขยย fails. Trio \( A,B,C \): only \( AB \) is close รขยย only 1 close pair รขยย fails.
๐ฐ So star with 3 edges: only 3 close pairs, but no trio has exactly 2.
๐ฐ Case 2: The three close pairs form a path: \( AB, BC, CD \). Edges: AB, BC, CD. Trio \( A,B,C \): AB, BC close รขยย 2 close pairs. Trio \( B,C,D \): BC, CD close รขยย 2 close pairs. Trio \( A,B,D \): AB, BD? BD not close, AD not close รขยย only AB รขยย 1 close รขยย fails. Trio \( A,C,D \): AC not close รขยย 0 รขยย fails. So only the two middle trios (A,B,C) and (B,C,D) each have exactly 2 close pairs.
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