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Lemma 55.5.9. Classification of proper subgraphs of the form

\[ \xymatrix{ \bullet \ar@{-}[r] & \bullet \ar@{..}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] \ar@{-}[d] & \bullet \\ & & & \bullet } \]

Let $t > 4$ and $n > t + 1$. Then given $t + 1$ distinct $(-2)$-indices $i_1, \ldots , i_{t + 1}$ such that $a_{i_ ji_{j + 1}}$ is nonzero for $j = 1, \ldots , t - 1$ and $a_{i_{t - 1}i_{t + 1}}$ is nonzero, then we have the $a$'s and $w$'s

  1. are given by $w_{i_1} = w_{i_2} = \ldots = w_{i_{t + 1}} = w$, $a_{i_ ji_{j + 1}} = w$ for $j = 1, \ldots , t - 1$, $a_{i_{t - 1}i_{t + 1}} = w$ and $a_{i_ ji_ k} = 0$ for other pairs $(j, k)$ with $j > k$.

Proof. See discussion above. $\square$

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