We study the phase-diagram of a parity and time-reversal (PT ) symmetric tight-binding chain with N sites and hopping energy J, in the presence of two impurities with imaginary potentials ±iγ located at arbitrary (P-symmetric) positions (m,m = N + 1 − m) on the chain where m ≤ N/2.We find that except in the two special cases where impurities are either the farthest or the closest, the PT -symmetric region -defined as the region in which all energy eigenvalues are real -is algebraically fragile. We analytically and numerically obtain the critical impurity potential γ P T and show that γ P T ∝ 1/N → 0 as N → ∞ except in the two special cases. When the PT symmetry is spontaneously broken, we find that the maximum number of complex eigenvalues is given by 2m. When the two impurities are the closest, we show that the critical impurity strength γ P T in the limit N → ∞ approaches J (J/2) provided that N is even (odd). For an even N the PT symmetry is maximally broken whereas for an odd N , it is sequentially broken. Our results show that the phase-diagram of a PT -symmetric tight-binding chain is extremely rich and that, in the continuum limit, this model may give rise to new PT -symmetric Hamiltonians. * yojoglek@iupui.edu