On the geodeticity of the contour of a graph

The eccentricity of a vertex v in a graph G is the maximum distance of v from any other vertex of G and v is a contour vertex of G if each vertex adjacent to v has eccentricity not greater than the eccentricity of v. The set of contour vertices of G is geodetic if every vertex of G lies on a shortest path between a pair of contour vertices. In this paper, we firstly investigate the existence of operations on graphs that allow to construct graphs in which the contour is geodetic. Then, after providing an alternative proof of the fact that the contour is geodetic in every HHD-free graph, we show that the contour is geodetic in every cactus and in every graph whose blocks are HHD-free or cycles or cographs. Finally, we generalize the above result by introducing the concept of geodetic-contour-preserving class of graphs and by proving that, if each block B in a graph G belongs to a class g,B of graphs which is geodetic-contour-preserving, then the contour of G is geodetic. (C) 2014 Elsevier B.V. All rights reserved.