The Decomposition of Graphs with Extended Cyclic Triples

In this paper, we study the decomposition of the graph \( (\lambda D_v)^{+\alpha} \) into extended cyclic triples, for all \( \lambda \geq \alpha \). By an extended cyclic triple, we mean a loop, a loop with symmetric arcs attached (known as a lollipop), or a directed \( 3 \)-cycle (known as a cyclic triple).