  Non-Hermitian, tight-binding $\mathcal{PT}$-symmetric models are extensively
studied in the literature. Here, we investigate two forms of non-Hermitian
Hamiltonians to study the $\mathcal{PT}$-symmetry breaking thresholds and
features of corresponding surfaces of exceptional points (EPs). They include
one-dimensional chains with uniform or 2-periodic tunnelling amplitudes, one
pair of balanced gain and loss potentials $\Delta\pm\i\gamma$ at
parity-symmetric sites, and periodic or open boundary conditions. By
introducing a Hermitian detuning potential, we obtain the dependence of the
$\mathcal{PT}$-threshold, and therefore the exceptional-point curves, in the
parameter space of detuning and gain-loss strength. By considering several such
examples, we show that EP curves of a given order generically have cusp-points
where the order of the EP increases by one. In several cases, we obtain
explicit analytical expressions for positive-definite intertwining operators
that can be used to construct a complex extension of quantum theory by
re-defining the inner product. Taken together, our results provide a detailed
understanding of detuned tight-binding models with a pair of gain-loss
potentials.
