In the Kidney Exchange Problem (KEP), we consider a pool of altruistic donors and incompatible patient-donor pairs. 
Kidney exchanges can be modelled in a directed weighted graph as circuits starting and ending in an incompatible pair or as paths starting at an altruistic donor.
The weights on the arcs represent the medical benefit which measures the quality of the associated transplantation.
For medical reasons, circuits and paths are of limited length and are associated with a medical benefit to perform the transplants.
The aim of the KEP is to determine a set of disjoint kidney exchanges of maximal medical benefit or maximal cardinality (all weights equal to one).
In this work, we consider a type of uncertainty in the KEP which stem from from the failure of a transplantation (existence of the arcs).
The uncertainty is modelled via an uncertainty set with constant budget.
The robust approach entails finding the best KEP solution in the worst-case scenario within the uncertainty set.
We modelled the robust counter-part by means of a max-min formulation which is defined on exponentially-many variables associated with the circuits and paths.
We introduce a representative-based reformulation for the max-min formulation and propose an exact approach to solve it.
The core algorithm is based on a Branch-Price-and-Cut approach where the exponentially-many variables are dynamically generated.
The computational experiments prove the efficiency of our approach against the literature.