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Nowadays the isogeometric analysis (IGA) represents an innovative method that merges design and numerical computations into a unified formulation. In such a context we apply the isogeometric concept based on T-splines and Non Uniform Rational B-Splines (NURBS) discretizations to study the interfacial contact and debonding problems between deformable bodies in large deformations. More in detail, we develop and test a generalized large deformation contact algorithm which accounts for both frictional contact and mixed-mode cohesive debonding in a unified setting. Some numerical examples are provided for varying resolutions of the contact and/or cohesive zone, which show the accuracy of the solutions and demonstrate the potential of the method to solve challenging 2D contact and debonding problems. The superior accuracy of T-splines with respect to NURBS interpolations for a given number of degrees of freedom (Dofs) is always proved. [ABSTRACT FROM AUTHOR]