  Tension-continuous (shortly TT) mappings are mappings between the edge sets
of graphs. They generalize graph homomorphisms. From another perspective,
tension-continuous mappings are dual to the notion of flow-continuous mappings
and the context of nowhere-zero flows motivates several questions considered in
this paper.
  Extending our earlier research we define new constructions and operations for
graphs (such as graphs Delta(G)) and give evidence for the complex relationship
of homomorphisms and TT mappings. Particularly, solving an open problem, we
display pairs of TT-comparable and homomorphism-incomparable graphs with
arbitrarily high connectivity.
  We give a new (and more direct) proof of density of TT order and study graphs
such that TT mappings and homomorphisms from them coincide; we call such graphs
homotens. We show that most graphs are homotens, on the other hand every vertex
of a nontrivial homotens graph is contained in a triangle. This provides a
justification for our construction of homotens graphs.
