New Results on the Tree Graph Defined by a Set of Cycles

For a set C(G) 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 U S contains exactly one cycle and this cycle lies in C(G). For some special graphs we find some minimal cycle sets to keep T(G, C) connected. For any 2-connected graph G, we find some cycle sets to maintain the connectedness of T(G, C).