Considering the bounded-degree graph model, we show that if the degree bound is two, then every graph property can be tested within query complexity that only depends on the proximity parameter. Specifically, the query complexity is \(\textrm{poly}(1/\epsilon )\) , where \(\epsilon \) denotes the proximity parameter. The key observation is that a graph of maximum degree two consists of a collection of paths and cycles, and that a collection of long paths and long cycles is relatively close (in this model) to a single cycle.