netscomplements

Netscomplements is a term that appears in discussions of nets in mathematics and network theory to denote a complementary construction associated with a given net or network. There is no single, universally accepted definition; rather, the phrase is used to describe different objects depending on the context.

In graph theory and network analysis, a common interpretation treats a net as a simple graph G = (V, E). The netscomplement G^c is the graph with the same vertex set V and edge set E^c consisting of all non-edges: E^c = { {u, v} : u ≠ v and {u, v} ∉ E }. The netscomplement thus contains every pair of vertices that is not connected in G. For directed networks, a directed netscomplement is defined similarly, using the non-edge relations that preserve direction.

In topology and the study of nets, the term is used in several nonstandard ways, reflecting the same idea of a complement relative to a chosen object. Common nonstandard definitions include (a) the complement of the net’s image in the ambient space, X \ im(N); (b) the complement of the net’s closure, X \ cl(im(N)); or (c) the set of points that are not accumulation points of the net. Because there is no single convention, authors specifying the precise meaning when the term is used is important.

History and usage: Netscomplements is found mainly in niche or didactic discussions as a convenient shorthand for complementary constructions related to nets or networks. See also nets, graph complement, and topological complements for related ideas.