An Isograft is a graft

Medical grafting

Grafting refers to a surgical procedure to move tissue from one site to another on the body, or from another person, without bringing its own blood supply with it. Instead, a new blood supply grows in after it is placed. A similar technique where tissue is transferred with the blood supply intact...

 of tissue between two individuals who are genetically identical (i.e. monozygotic twins). Transplant rejection

Transplant rejection

Transplant rejection occurs when transplanted tissue is rejected by the recipient's immune system, which destroys the transplanted tissue. Transplant rejection can be lessened by determining the molecular similitude between donor and recipient and by use of immunosuppressant drugs after...

 between two such individuals virtually never occurs.

As monozygotic twins have the same major histocompatibility complex

Major histocompatibility complex

Major histocompatibility complex is a cell surface molecule encoded by a large gene family in all vertebrates. MHC molecules mediate interactions of leukocytes, also called white blood cells , which are immune cells, with other leukocytes or body cells...

, there is very rarely any rejection of transplanted tissue by the adaptive immune system

Adaptive immune system

The adaptive immune system is composed of highly specialized, systemic cells and processes that eliminate or prevent pathogenic growth. Thought to have arisen in the first jawed vertebrates, the adaptive or "specific" immune system is activated by the “non-specific” and evolutionarily older innate...

. Furthermore, there is virtually no incidence of graft-versus-host disease. This forms the basis for why the preferred choice of a physician considering an organ donor will be a monozygotic twin.