We have already definitions for open and closed neighborhoods of a vertex, which differs only in the fact that the first never contains the vertex, and the latter always contains the vertex. One of these definitions, however, cannot be simply derived from the other. This would be possible if a definition of a semiclosed neighborhood was available, see dfsclnbgr2. The definitions for open and closed neighborhoods could be derived from such a more simple, but otherwise probably useless definition, see dfnbgr5 and dfclnbgr5. Depending on the existence of certain edges, a vertex belongs to its semiclosed neighborhood or not.
An alternate approach is to introduce semiopen neighborhoods, see dfvopnbgr2. The definitions for open and closed neighborhoods could also be derived from such a definition, see dfnbgr6 and dfclnbgr6. Like with semiclosed neighborhood, depending on the existence of certain edges, a vertex belongs to its semiopen neighborhood or not.
It is unclear if either definition is/will be useful, and in contrast to dfsclnbgr2, the definition of semiopen neighborhoods is much more complex.