Abstract
A graphG isconvergent when there is some finite integern ≥ 0, such that then-th iterated clique graphK n(G) has only one vertex. The smallest suchn is theindex ofG. TheHelly defect of a convergent graph is the smallestn such thatK 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 difference of its index and diameter by more than one. In the present paper an affirmative answer to the question is given. For any arbitrary finite integern, a graph is exhibited, the Helly defect of which exceeds byn the difference of its index and diameter.
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Bornstein, C.F., Szwarcfiter, J.L. On clique convergent graphs. Graphs and Combinatorics 11, 213–220 (1995). https://doi.org/10.1007/BF01793007
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DOI: https://doi.org/10.1007/BF01793007