Abstract
As a variant of the famous graph reconstruction problem we characterize classes of graphs of order n such that all their induced subgraphs on k ⩽ n vertices satisfy some property related to the number of edges or to the vertex degrees. We give complete solutions for the properties (i) to be regular, (ii) to be regular modulo m ⩾ 2 or (iii) to have one of two possible numbers of edges. Furthermore, for an order n large enough, we give solutions for the properties (iv) to be bi-regular or (v) to have a bounded difference between the maximum and the minimum degree.
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