2.3 Mapping cyberspace over a visualization media

### 2.3.2 Morphisms

To give a proper view of the structure of cyberspace over a medium, various requirements for the mapping are described below, in term of morphisms and from weak to strong constraints.
An exact match from the set to set is a mapping that has a corresponding relation tuple (element) in for each relation in . A mapping fulfilling this requirement is called a homomorphism from to such that

.

Consider that a mapping exists wherein there is a one-to-one correspondence between the vertices in and the vertices in a subgraph of such that a pair of vertices are adjacent in if and only if the corresponding pair of vertices are adjacent in the subgraph of . This is in fact the condition for a monomorphic mapping:

which can also be described as a isomorphic mapping to a subset from the relation such that

.

Note that all these morphism problems have been shown to be NP-complete and therefore cannot be solved exactly in polynomial time.

Cyberspace geography visualization - 15 October 1995

Luc Girardin, The Graduate Institute of International Studies