A Necessary and Sufficient Condition on the Tree Graph Defined by a Set of Cycles

For a set C(G) of cycles of a connected graph G, we defined T(G, C) as the graph with one vertex for each spanning tree of G, in which two trees R and S are adjacent if R υ S contains exactly one cycle and this cycle lies in C(G). For any 2-connected graph G, we give a necessary and sufficient condition for T(G, C) to be connected. And some cycle sets to maintain the connectedness of T(G, C) are also discussed.