Constructing and classifying neighborhood anti-Sperner graphs

For a simple graph G let NG(u) be the (open) neighborhood of vertex u V (G). Then G is neighborhood anti-Sperner (NAS) if for every u there is a v V (G)\{u} with NG(u) NG(v). And a graph H is neighborhood distinct (ND) if every neighborhood is distinct, i.e., if NH(u) = NH(v) when u = v, for all u and v V (H). In Porter and Yucas [3] 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 r-regular NAS graphs for r = 0, 1, . . ., 6.

• Type: Journal
• Year: 2008
• Where: DM
• Authors: John P. McSorley
• Keywords: DM 2008, Null Graphs, Regular Nas Graph, Simple Graph