On the existence problem of the total domination vertex critical graphs Original Research Article

The existence problem of the total domination vertex critical graphs has been studied in a series of articles. We first settle the existence problem with respect to the parities of the total domination number image and the maximum degree image: for even image except image, there is no image-image-critical graph regardless of the parity of image; for image or odd image and for even image, an image-image-critical graph exists if and only if image; for image or odd image and for odd image, if image, then image-image-critical graphs exist, if image, then image-image-critical graphs do not exist. The only remaining open cases are image, image. Second, we study these remaining open cases when image or odd image. As the previously known result for image, we also show that for image, there is no image-image-critical graph of order image. On the contrary, it is shown that for odd image there exists an image-image-critical graph for all image.