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📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 So, to count such trios, we count the number of triples where two edges form a path of length 2 (i.e., a "V" shape), and the third edge is absent.

📌 We identify all such "V-shaped" trios in the graph with 5 nodes and 3 close pairs.

📌 Let the 3 close pairs form a path: \( A - B - C \). Then the pairs are \( AB, BC, AC \), with only \( AB \) and \( BC \) close â so 2 close pairs, 1 non-close (namely \( AC \)). This trio counts.

📌 Any other configuration of 3 close pairs that does not form such a path?

📌 Try a star: \( AB, AC, BC \) â but thatâs 3 edges all connected to B â actually forms a triangle. Then every pair is close: 3 close pairs â violates our "exactly 2" condition.

📌 Can we have a non-path configuration with exactly 2 close pairs?

📌 Example: close pairs: \( AB, CD, DE \). Then the trio \( B,C,D \): pairs \( BC \) (not close), \( CD, DE \) â only 2 close: \( CD, DE \). So this trio has exactly 2 close pairs and 1 non-close.

📌 But this is not a path of length 2 â itâs a chain of 3.

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