Geodesic
On the tree graph of a connected graph
Discussiones Mathematicae. Graph Theory, Tome 28 (2008) no. 3, pp. 501-510

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Let G be a graph and C be a set of cycles of G. The tree graph of G defined by C, is the graph T(G,C) that has one vertex for each spanning tree of G, in which two trees T and T' are adjacent if their symmetric difference consists of two edges and the unique cycle contained in T ∪ T' is an element of C. We give a necessary and sufficient condition for this graph to be connected for the case where every edge of G belongs to at most two cycles in C.
Keywords: tree graph, property Δ*, property Δ⁺ 
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Figueroa, Ana; Rivera-Campo, Eduardo. On the tree graph of a connected graph. Discussiones Mathematicae. Graph Theory, Tome 28 (2008) no. 3, pp. 501-510. http://geodesic.mathdoc.fr/item/DMGT_2008_28_3_a9/