A stent graft and method for positioning and deploying the stent graft within a vessel system that includes a main vessel and a branch vessel emanating from the main vessel. The stent graft includes a tubular shaped main body formed from graft material, a branch opening (aperture)(ring) in the graft material of the main body whose position can be varied. A tubular shaped branch graft can extend from the main graft. A side wall of the main body may be configured as a series of connected annular corrugations or pleats, and coupled to the main body to define and provide variable positioning of its branch opening (aperture).