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difference between a watch and a warning - IQnection

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 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.

📌 But we fix the set of 3 close pairs. We need among all 3-edge sets with exactly two edges forming a path of length 2, how many such sets exist.

📌 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.

📌 Let \( G \) be the graph with 5 vertices and 3 edges. Let \( d(v) \) be degree of \( v \).

📌 A path of length 2 is determined by a middle vertex and two neighbors.

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