  We investigate the Bell inequalities derived from the graph states with
violations detectable even with the presence of noises, which generalizes the
idea of error-correcting Bell inequalities [Phys. Rev. Lett. 101, 080501
(2008)]. Firstly we construct a family of valid Bell inequalities tolerating
arbitrary $t$-qubit errors involving $3(t+1)$ qubits, e.g., 6 qubits suffice to
tolerate single qubit errors. Secondly we construct also a
single-error-tolerating Bell inequality with a violation that increases
exponentially with the number of qubits. Exhaustive computer search for optimal
error-tolerating Bell inequalities based on graph states on no more than 10
qubits shows that our constructions are optimal for single- and double-error
tolerance.
