On clique convergent graphs

A graph G is convergent when there is some nite integer n 0, such that the n-th iterated clique graph K n (G) has only one vertex. The smallest such n is the index of G. The Helly defect of a convergent graph is the smallest n such that K n (G) is clique Helly, that is, its maximal cliques satisfy the Helly property. Bandelt and Prisner proved that the Helly defect of a chordal graph is at most one and asked whether there is a graph whose Helly defect exceeds the diierence of its index and diameter by more than one. In the present paper an aarmative answer to the question is given. For any arbitrary nite integer n, a graph is exhibited, the Helly defect of which exceeds by n the diierence of its index and diameter.