  This paper mainly investigates the strong convergence and stability of the
truncated Euler-Maruyama (EM) method for stochastic differential delay
equations with variable delay whose coefficients can be growing super-linearly.
By constructing appropriate truncated functions to control the super-linear
growth of the original coefficients, we present a type of the truncated EM
method for such SDDEs with variable delay, which is proposed to be approximated
by the value taken at the nearest grid points on the left of the delayed
argument. The strong convergence result (without order) of the method is
established under the local Lipschitz plus generalized Khasminskii-type
conditions and the optimal strong convergence order $1/2$ can be obtained if
the global monotonicity with U function and polynomial growth conditions are
added to the assumptions. Moreover, the partially truncated EM method is proved
to preserve the mean-square and H_\infty stabilities of the true solutions.
Compared with the known results on the truncated EM method for SDDEs, a better
order of strong convergence is obtained under more relaxing conditions on the
coefficients, and more refined technical estimates are developed so as to
overcome the challenges arising due to variable delay. Lastly, some numerical
examples are utilized to confirm the effectiveness of the theoretical results.
