Geodesic
Decompositions into two paths
Discussiones Mathematicae. Graph Theory, Tome 25 (2005) no. 3, pp. 325-329

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It is proved that a connected multigraph G which is the union of two edge-disjoint paths has another decomposition into two paths with the same set, U, of endvertices provided that the multigraph is neither a path nor cycle. Moreover, then the number of such decompositions is proved to be even unless the number is three, which occurs exactly if G is a tree homeomorphic with graph of either symbol + or ⊥. A multigraph on n vertices with exactly two traceable pairs is constructed for each n ≥ 3. The Thomason result on hamiltonian pairs is used and is proved to be sharp.
Keywords: graph, multigraph, path decomposition, hamiltonian decomposition, traceable 
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Skupień, Zdzisław. Decompositions into two paths. Discussiones Mathematicae. Graph Theory, Tome 25 (2005) no. 3, pp. 325-329. http://geodesic.mathdoc.fr/item/DMGT_2005_25_3_a10/