In this section, we first define the patients' chain with the kidney seller. To form the chain, a kidney seller is introduced from the medical centers, so that there is no personal financial transaction between pairs of patient-donor. As you can see in Figure 4, the seller donates his kidney to patient 1, the donor 1 to the second patient and donor 2 to the second patient, and the second donor to the third patient; the third donor also his kidney to the other chains. This chain, in such a way that if the patient i could not be in the chain, the transplantation is not possible for the patients i+1 to the end of the chain and it will be possible for the sub-chain
