= Computational-basis measurement of a graph-state vertex
= Graph-state vertex deletion
{synonym}
For a <graph state> $|G\rangle=\prod_{ij\in E}CZ_{ij}|+\rangle^{\otimes|V|}$, measuring <vertex> $v$ in the <computational basis> with result $r$ gives the normalized state
$$
\left(\prod_{u\in N(v)}Z_u^r\right)|G-v\rangle.
$$
Each <Controlled-Z gate> from $v$ to a neighbour acts as $Z_u^r$ after projecting $v$ onto $|r\rangle$; all other <edges> remain unchanged. The projection contributes $1/\sqrt2$, so either result has <probability> $1/2$. For a four-cycle, deleting one <vertex> leaves a path and adds byproducts on its two endpoints.
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