The connectivities of trunk graphs of 2-connected graphs

Let G be a connected graph and V* set of all the spanning trees except stars in G. An edge in a spanning tree is called 'inner' if the edge is not incident to endvertices. Define an adjacency relation in V* as follows; two spanning trees t(1) and t(2) is an element of V* are called to be adjacent if there exist inner edges e(i) is an element of E(t(i)) such that t(1) - e(1) = t(2) - e(2). The resultant graph is a subgraph of the tree graph, and we call it simply a trunk graph. The purpose of this paper is to show that if a 2-connected graph with at least five vertices is k-edge connected, then its trunk graph is (k - 1)-connected.