The forcing hull and forcing geodetic numbers of graphs Original Research Article

For every pair of vertices image in a graph, a image geodesic is a shortest path from image to image. For a graph image, let image denote the set of all vertices lying on a image geodesic. Let image and image denote the union of all image for all image. A subset image is a convex set of image if image. A convex hull image of image is a minimum convex set containing image. A subset image of image is a hull set of image if image. The hull number image of a graph image is the minimum cardinality of a hull set in image. A subset image of image is a geodetic set if image. The geodetic number image of a graph image is the minimum cardinality of a geodetic set in image. A subset image is called a forcing hull (or geodetic) subset of image if there exists a unique minimum hull (or geodetic) set containing image. The cardinality of a minimum forcing hull subset in image is called the forcing hull number image of image and the cardinality of a minimum forcing geodetic subset in image is called the forcing geodetic number image of image. In the paper, we construct some 2-connected graph image with image, or image, and prove that, for any nonnegative integers image, image, and image with image, there exists a 2-connected graph image with image or image. These results confirm a conjecture of Chartrand and Zhang proposed in [G. Chartrand, P. Zhang, The forcing hull number of a graph, J. Combin. Math. Combin. Comput. 36 (2001) 81–94].