new vertex connected to the two vertices originally joined by that edge.\
– Client is given an MBQC measurement pattern $\bbbm_{\textrm{Comp}}$ which when applied to the dotted graph state $\ket{D(G)}$ performs the desired computation, in a fault-tolerant way that can detect or correct errors fewer than $\delta/2$.\
– Client generates the dotted triple-graph $DT(G)$ and selects a trap-colouring according to definition \[trap colouring\], which is done by choosing independently the colours for each set $P_v$.\
– Client for all red vertices will send dummy qubits and thus perform break operation.\
– Client chooses the green graph to perform the computation.\
– Client for the white graph will send dummy qubits for all added qubits $a^{e}_w$ and thus generate white isolated qubits at each primary vertex set $P_{v}$. Similarly for the black graph the client will send dummy qubits for the primary qubits $p^v_b$ and thus generate black isolated qubits at each added vertex set $A_{e}$.\
– The set $D$ of the positions of dummy qubits is chosen as defined above (fixed by the trap-colouring).\
– A binary string $\mathbf{s}$ of length at most $3N(3c+1)$ represents the measurement outcomes. It is initially set to all zeros.\
– A sequence of measurement 