ngle}$. Multiple index $\delta$ on the other hand accounts for all states ${| {g_{(\mathcal{V}/\mathcal{Y})}}_\delta \rangle}$ generated from $Z$ or identity operators on the state ${| {g_{(\mathcal{V}/\mathcal{Y})}}_0 \rangle}\equiv\bigotimes_{i
\in\mathcal{V}/\mathcal{Y}} {| +_i \rangle}$ of the non-boundary qubits $\mathcal{V}/\mathcal{Y}$. Probability $\tilde{p}_{\gamma,\delta}$ is defined as the sum of all $p_{\mu}$ such that $\tilde{K}_\mu{| {G_{(\mathcal{Y},\mathcal{X})}}_0 \rangle}\otimes
{| {g_{(\mathcal{V}/\mathcal{Y})}}_0 \rangle}={| {G_{(\mathcal{Y},\mathcal{X})}}_\gamma \rangle}\otimes{| {g_{(\mathcal{V}/\mathcal{Y})}}_\delta \rangle}$. Because the $CZ$ gates explicitly factored out in state are local unitary operations with respect to the partition of interest, the entanglement of $\rho_t$, $E(\rho_t)$, reads $$\begin{aligned}
\label{ent}
E\big(\sum_{\gamma,\delta}
\tilde{p}_{\gamma,\delta}{| {G_{(\mathcal{Y},\mathcal{X})}}_\gamma \rangle}
{\langle {G_{(\mathcal{Y},\mathcal{X})}}_\gamma |}\otimes{| {g_{(\mathcal{V}/\mathcal{Y})}}_\delta \rangle}{\langle {g_{(\mathcal{V}/\mathcal{Y})}}_\delta |}\big).
 \end{aligned}$$ where $E$ is any convex entanglement quantifier not increasing under LOCC. In what follows, we first establish a lower and upper bound to this expression and then show that these bounds coincide, obtaining the exact expression of the graph-state entanglement evolution.

First, consider an LOCC protocol consisting of measuring all the non-boundary qubits $\mathcal{V}/\mathcal{Y}$ of the state within brackets in Eq. in the product basis composed by all orthonormal states $\{{| {g_{(\mathcal{V}/\mathcal{Y})}}_\delta \rangle}\}$ and tracing out the measured subsystem after communicating the outcomes. The remaining subsystem $\mathcal{Y}$ is [*flagged*]{} by each measurement outcome $\delta$ – meaning that outcome $\delta$ provides full information about to which state $\mathcal{Y}$ has been projected after each measurement run. The final entanglement after the entire protocol is then give