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1, show that this bound on the radius is best possible. Two results of Chang and Nemhauser concerning diameters and radii of chordal graphs are also corollaries.
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doi:10.1016/0012-365X(87)90099-9 
Copyright © 1987 Published by Elsevier Science B.V.
On local convexity in graphs
Received 26 August 1985;
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Abstract
A set K of nodes of a graph G is geodesically convex (respectively, monophonically convex) if K contains every node on every shortest (respectively, chordless) path joining nodes in K. We investigate the classes of graphs which are characterized by certain local convexity conditions with respect to geodesic convexity, in particular, those graphs in which balls around nodes are convex, and those graphs in which neighborhoods of convex sets are convex. For monophonic convexity, these conditions are known to be equivalent, and hold if and only if the graph is chordal. Although these conditions are not equivalent for geodesic convexity, each defines a generalization of the class of chordal graphs. A persistent theme here will be the analogies between these graphs and chordal graphs.
