Upper bounds on the image-domination number and the image-Roman domination number Original Research Article

Let image be a graph and let image be a positive integer. A subset image of the vertex set of image is a image-dominating set if every vertex not in image has at least image neighbors in image. The image-domination number image is the minimum cardinality among the image-dominating sets of image. A Roman image-dominating function on image is a function image such that every vertex image for which image is adjacent to at least image vertices image with image for image. The weight of a Roman image-dominating function is the value image. The minimum weight of a Roman image-dominating function on a graph image is called the Roman image-domination number image.