Cycle-continuous mappings – order structure

Given two graphs, a mapping between their edge-sets is cycle-continuous, if the preimage of every cycle is a cycle. Answering a question of DeVos, Nešetřil, and Raspaud, we prove that there exists an infinite set of graphs with no cycle-continuous mapping between them. Further extending this result, we show that ev-ery countable poset can be represented by graphs and existence of cycle-continuous mappings between them.