Balanced and strongly balanced Pk-designs

Given a graph $G$, a G-decomposition of the complete graph $K_v$ is a set of graphs, all isomorphic to $G$, whose edge sets partition the edge set of $K_v$. A $G$-decomposition of $K_v$ is also called a G-design and the graphs of the partition are said to be the blocks. A $G$-design is said to be balanced if the number of blocks containing any given vertex of $K_v$ is a constant.In this paper the concept of strongly balanced G-design is introduced and strongly balanced path-designs are studied. Furthermore, we determine the spectrum of those path-designs which are balanced, but not strongly balanced.