  Synchronization in systems of coupled Kuramoto oscillators may depend on
their natural frequencies, coupling, and underlying networks. In this paper, we
reduce the alternatives to only one by considering identical oscillators where
the only parameter that is allowed to change is the underlying network. While
such a model was analyzed over the past few decades by studying the size of the
basin of attraction of the synchronized state on restricted families of graphs,
here we address a qualitative question on general graphs. In an analogy to
resistive networks with current sources, we describe an algorithm that produces
initial conditions that are often outside of the basin of attraction of the
synchronized state. In particular, if a graph allows a cyclic graph clustering
with a sufficient number of clusters or contains a sufficiently long induced
subpath without cut vertices of the graph then there is a non-synchronous
stable phase-locked solution. Thus, we provide a partial answer to when the
synchronized state is not globally stable.
