Adaptive interface-Mesh un-Refinement (AiMuR) based Sharp-Interface Level-Set-Method (SILSM) is proposed for both uniform and non-uniform Cartesian-Grid. The AiMuR involves interface location based dynamic un-refinement (with merging of the four control volumes) of the Cartesian grid away from theinterface. The un-refinement is proposed for the interface solver only. A detailed numerical methodology is presented for the AiMuR and ghost-fluid method based SI-LSM. Advantage of the novel as compared to the traditional SI-LSM is demonstrated with a detailed qualitative as well as quantitative performance study, involving the SI-LSMs on both coarse grid and fine grid, for three sufficiently different two-phase flow problems:dam break, breakup of a liquid jet and drop coalescence. A superior performance of AiMuR based SI-LSM is demonstrated - the AiMuR on a coarser non-uniform grid (NU_c^AiMuR) is almost as accurate as the traditional SILSM on a uniform fine grid (U_ƒ) and takes a computational time almost same as that by the traditional SI-LSM on a uniform coarse grid (U_c). The AMuR is different from the existing Adaptive Mesh Refinement (AMR) asthe former involves only mesh un-refinement while the later involves both refinement and un-refinement of the mesh. Moreover, the proposed computational development is significant since the present adaptive un-refinement strategy is much simpler to implement as compared to that for the commonly used adaptive refinement strategies. The proposed numerical development can be extended to various other multi-physics, multi-disciplinaryand multi-scale problems involving interfaces.