Graphs with tranches: consequences for topology and dynamics

In this paper we compare quasi graphs and generalized $\sin(1/x)$-type continua - two classes that generalize topological graphs and contain the Warsaw circle as a non-trivial element. We show that neither class is the subset of the other, provide the characterization and present illustrative examples. We unify both approaches by considering the class of tranched graphs, connect it with objects found in literature and describe how the topological structure of its elements restricts possible dynamics.