Bridged Graphs and Geodisic Convexity

A graph G is bridged if each cycle C of length at least four contains two vertices whose distance from each other in G is strictly less than that in C. The class of bridged graphs is an extension of the class of chordal (or triangulated) graphs which arises in the study of convexity in graphs. A set K of vertices of a graph G is geodesically convex if K contains every vertex or every shortest path joining vertices in K. In another paper by R.E. Jamison and the author, it is shown that a graph is bridged if and only if the closed neighborhood of every geodesically convex set is again geodesically convex. This paper contains several results concerning geodesically convex sets in bridged graphs. As an interesting consequence of these results we obtain a recursive characterization of the class of bridged graphs.