A counterexample to a result on the tree graph of a graph

Given a set of cycles C of a graph G, the tree graph of G, defined by C, is the graph T (G,C) whose vertices are the spanning trees of G and in which two trees R and S are adjacent if R∪S contains exactly one cycle and this cycle lies in C. Li et al. [Discrete Math. 271 (2003), 303–310] proved that if the graph T (G,C) is connected, then C cyclically spans the cycle space of G. Later, Yumei Hu [Proc. 6th Int. Conf. Wireless Communications Networking and Mobile Comput. (2010), 1–3] proved that if C is an arboreal family of cycles of G which cyclically spans the cycle space of a 2-connected graph G, then T (G,C) is connected. In this note we present an infinite family of counterexamples to Hu’s result.