Abstract
Consider two tournaments T and T' on the same vertex set V, with vertical bar V vertical bar = n >= 2 and let k be a positive integer. The tournaments T and T' are {-k}-hypomorphic (resp. k-hypomorphic), whenever for every subset X of V with vertical bar X vertical bar = n - k (resp. vertical bar X vertical bar = k), the subtournaments T' [X] and T [X] are isomorphic. The tournament T is {-k}-self dual (resp. k-self dual) if it is {-k}-hypomorphic (resp. k-hypomorphic) to its dual. In this work, we firstly characterize the pairs of {-3}-hypomorphic tournaments, following our previous study of "the {-3}-reconstruction and the {-3}-self duality of tournaments", and then we characterize the pairs of k-hypomorphic tournaments for k >= 1.