An endovascular graft is provided that includes a stent structure adapted to move between a collapsed and a deployed configuration. An endovascular graft material is mounted with respect to the stent structure. At least one lumen- or chute-forming structure is associated with the stent structure internal to the endovascular graft material. The at least one lumen- or chute-forming structure is adapted to move between a first position and a second position, wherein the at least one lumen- or chute-forming structure is rolled within itself in the first position, and wherein the at least one lumen- or chute-forming structure moves from the first position to the second position by unrolling. When in the second position, the at least one lumen- or chute-forming structure defines a passage having a greater cross-sectional area as compared to the first position. Stent-graft deployment to one or more branch arteries/vessels may be accomplished through the at least one lumen- or chute-forming structure. The present disclosure also provides advantageous methods for deploying the disclosed endovascular graft and delivery of stent-grafts to branch arteries/vessels therethrough.
