Transformation of Spanning Trees in a \(2\)-Connected Graph

Let \(T\) be a spanning tree of a graph \(G\). This paper is concerned with the following operation: we remove an edge \(e \in E(T)\) from \(T\), and then add an edge \(f \in E(G) – E(T)\) so that \(T – e + f\) is a spanning tree of \(G\). We refer to this operation of obtaining \(T – e + f\) from \(T\) as the transfer of \(e\) to \(f\). We prove that if \(G\) is a \(2\)-connected graph with \(|V(G)| \geq 5\), and if \(T_1\) and \(T_2\) are spanning trees of \(G\) which are not stars, then \(T_1\) can be transformed into \(T_2\) by repeated applications of a transfer of a nonpendant edge (an edge \(xy\) of a tree \(T\) is called a nonpendant edge of \(T\) if both of \(x\) and \(y\) have degree at least \(2\) in \(T\)).