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A sum graph G is a graph with a mapping of the vertex set of G onto a set of positive integers S in such a way that two vertices of G are adjacent if and only if the sum of their labels is an element of S . In an exclusive sum graph the integers of S that are the sum of two other integers of S form a set of integers that label a collection of isolated vertices associated with the graph G . A graph bears a k-exclusive sum labelling (abbreviated k -ESL), if the set of isolated vertices is of cardinality k . In this paper, observing that the property of having a k -ESL is hereditary, we provide a characterisation of graphs that have a k -exclusive sum labelling, for any k ≥ 1 , in terms of describing a universal graph for the property. [ABSTRACT FROM AUTHOR]