The presence of non-standard interaction (NSI) in neutrino physics motivates us to investigate the two solutions for solar neutrino problem, one is for standard Large Mixing Angle (LMA) with solution \(\sin ^2{\theta _{12}}=0.3\) and other is \(\sin ^2{\theta _{12}}=0.7\) . The latter is known as the Dark-Large Mixing Angle (D-LMA) solution. In this work, we have investigated the one zero texture and vanishing trace of inverse neutrino mass matrix in the context of LMA and D-LMA solution of the solar neutrino problem. We have six cases to study and we found that the textures which are allowed for LMA solution are also allowed for D-LMA solution. Further, we show the correlations between different parameters for \(3\sigma \) ranges of neutrino oscillation data. The case \(m_{11}^{-1}\) is not allowed as it cannot reproduce the correct neutrino phenomenology. Similarly, the case with \(m_{12}^{-1}\) predicts normal hierarchical neutrino masses and is consistent with both LMA and D-LMA phenomenology.