H-Decompositions of r-graphs when H is an r-graph with exactly 2 edges

Given two r-graphs G and H, an H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a graph isomorphic to H. The minimum number of parts in an H-decomposition of G is denoted by φH(G). By a 2-edge-decomposition of an r-graph we mean an H-decomposition for any fixed r-graph H with exactly 2 edges. In the special case where the two edges of H intersect in exactly 1, 2 or r − 1 vertices these 2-edge-decompositions will be called bowtie, domino and kite respectively. The value of the function φrH(n) will be obtained for bowtie, domino and kite decompositons of r-graphs.