On local convexity in graphs

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.