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- Case 2: The three close pairs form a path: \( AB, BC, CD \). Edges: AB, BC, CD. Trio \( A,B,C \): AB, BC close â 2 close pairs. Trio \( B,C,D \): BC, CD close â 2 close pairs. Trio \( A,B,D \): AB, BD? BD not close, AD not close â only AB â 1 close â fails. Trio \( A,C,D \): AC not close â 0 â fails. So only the two middle trios (A,B,C) and (B,C,D) each have exactly 2 close pairs.
- So one two-edge path with distinct middle vertex.
- Number of such paths: a path of 3 nodes has 2 endpoints and one middle. Number of such paths in a complete graph of 5 nodes: number of ways to choose 3 distinct nodes: \( inom{5}{3} = 10 \), each induces a path unless it's complete or linear.
- Alternate approach: total number of trios with exactly two close pairs.
- Each such trio corresponds to a 2-edge path (i.e., two adjacent edges sharing a vertex, not forming a triangle).
- For each internal node of degree at least 2 in the close pair graph, we can count how many such paths go through it.
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