  We consider mappings between edge sets of graphs that lift tensions to
tensions. Such mappings are called tension-continuous mappings (shortly TT
mappings). Existence of a TT mapping induces a (quasi)order on the class of
graphs, which seems to be an essential extension of the homomorphism order
(studied extensively, see [Hell-Nesetril]). In this paper we study the
relationship of the homomorphism and TT orders. We stress the similarities and
the differences in both deterministic and random setting. Particularly, we
prove that TT order is dense and universal and we solve a problem of M. DeVos
et al.
