Geodesic
Decompositions of multigraphs into parts with two edges
Discussiones Mathematicae. Graph Theory, Tome 22 (2002) no. 1, pp. 113-121

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Given a family of multigraphs without isolated vertices, a multigraph M is called -decomposable if M is an edge disjoint union of multigraphs each of which is isomorphic to a member of . We present necessary and sufficient conditions for the existence of such decompositions if comprises two multigraphs from the set consisting of a 2-cycle, a 2-matching and a path with two edges.
Keywords: edge decomposition, multigraph, line graph, 1-factor 
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Ivančo, Jaroslav; Meszka, Mariusz; Skupień, Zdzisław. Decompositions of multigraphs into parts with two edges. Discussiones Mathematicae. Graph Theory, Tome 22 (2002) no. 1, pp. 113-121. http://geodesic.mathdoc.fr/item/DMGT_2002_22_1_a9/