On weakly connected domination in graphs II Original Research Article

A dominating set image is a weakly connected dominating set of a connected graph image if image is connected. The weakly connected domination number of image, denoted image, is image is a weakly connected dominating set of image. We characterize graphs image for which image for every connected induced subgraph image of image, where image is the domination number of a graph. We provide a constructive characterization of trees image for which image. Lastly, we constructively characterize the trees image in which every vertex belongs to some weakly connected dominating set of cardinality image.