The upper connected geodetic number and forcing connected geodetic number of a graph Original Research Article

For a connected graph image of order image, a set image is a geodetic set of image if each vertex image lies on an image geodesic for some elements image and image in image. The minimum cardinality of a geodetic set of image is defined as the geodetic number of image, denoted by image. A geodetic set of cardinality image is called a image-set of image. A connected geodetic set of image is a geodetic set image such that the subgraph image induced by image is connected. The minimum cardinality of a connected geodetic set of image is the connected geodetic number of image and is denoted by image. A connected geodetic set of cardinality image is called a image-set of image. A connected geodetic set image in a connected graph image is called a minimal connected geodetic set if no proper subset of image is a connected geodetic set of image. The upper connected geodetic number image is the maximum cardinality of a minimal connected geodetic set of image. We determine bounds for image and determine the same for some special classes of graphs. For positive integers image and image with image, there exists a connected graph image with image, image and image. Also, for any positive integers image, there exists a connected graph image such that image, image and image. A subset image of a image-set image is called a forcing subset for image if image is the unique image-set containing image. A forcing subset for image of minimum cardinality is a minimum forcing subset of image. The forcing connected geodetic number of image, denoted by image, is the cardinality of a minimum forcing subset of image. The forcing connected geodetic number of image, denoted by image, is image, where the minimum is taken over all image-sets image in image. It is shown that for every pair image of integers with image, there exists a connected graph image such that image and image.