On tension-continuous mappings

Tension-continuous (shortly TT) mappings are mappings between the edge sets of graphs. They generalize graph homomorphisms. At the same time, tension-continuous mappings are a dual notion to 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 ΔM(G)ΔM(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.