= Equatorial measurement of a graph-state leaf
Let $1$ be a leaf with neighbour $2$ and let the graph on the other <vertices> be generated by the <Controlled-Z gate> product $E_{\mathrm{rest}}$. If the leaf has <Pauli Z gate> frame $Z_1^r$, measuring it at angle $\alpha$ with result $s$ applies <one-bit teleportation> and leaves
$$
E_{\mathrm{rest}}\left(X_2^sJ(\alpha)_2Z_2^r|+\rangle_2\otimes|+\rangle_{\mathrm{other}}\right).
$$
Any known <Pauli Z gate> frames on other <vertices> remain as additional factors before those <vertices>' input states. The identity $J(\alpha)Z^r=X^rJ(\alpha)$ combines the leaf byproduct with the measurement result.
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