Constructing and classifying neighborhood anti-Sperner graphs Original Research Article

For a simple graph image let image be the (open) neighborhood of vertex image. Then image is neighborhood anti-Sperner (NAS) if for every image there is a image with image. And a graph image is neighborhood distinct (ND) if every neighborhood is distinct, i.e., if image when image, for all image and image. In Porter and Yucas [T.D. Porter, J.L. Yucas. Graphs whose vertex-neighborhoods are anti-sperner, Bulletin of the Institute of Combinatorics and its Applications 44 (2005) 69–77] a characterization of regular NAS graphs was given: ‘each regular NAS graph can be obtained from a host graph by replacing vertices by null graphs of appropriate sizes, and then joining these null graphs in a prescribed manner’. We extend this characterization to all NAS graphs, and give algorithms to construct all NAS graphs from host ND graphs. Then we find and classify all connected image-regular NAS graphs for image.