B.D.Acharya and E. Sampathkumar  defined Graphoidal cover as partition of edge set of a graph G into internally disjoint paths (not necessarily open). The minimum cardinality of such cover is known as graphoidal covering number of G.
Bulletin of Mathematical Sciences and Applications (Volume 9)
A. N. Murugan "Magic Graphoidal on Class of Trees", Bulletin of Mathematical Sciences and Applications, Vol. 9, pp. 33-44, 2014