Summary: This paper deals with boundary value inverse problems on a metric graph, the structure of the graph being assumed unknown. The question under consideration is how to detect from the dynamical and/or spectral inverse data whether the graph contains cycles (is not a tree). For any graph \(\Omega \), the dynamical as well as spectral boundary inverse data determine the so-called wave diameter \(d_w: H^{-1}(\Omega ) \rightarrow {\mathbb R} \) defined on functionals supported in the graph. The known fact is that if \(\Omega \) is a tree then \(d_w\geq 0\) holds and, in this case, the inverse data determine \(\Omega \) up to isometry. A graph \(\Omega \) is said to be coordinate if the functions \(\{\text{dist}_{\Omega }(\cdot, \gamma )\}_{\gamma\in\partial \Omega }\) constitute a coordinate system on \(\Omega \). For such graphs, we propose a procedure, which reveals the presence/absence of cycles. The hypothesis is that \(\Omega\) contains cycles if and only if \(d_w\) takes negative values. We do not justify this hypothesis in the general case but reduce it to a certain special class of graphs (suns).