Suppose that $H$ is a graph on $V$ vertices, using half of the possible edges (like a a path of length 3, or a star on 4 vertices). We let $H^c$ be the complement of $H$ in $V$, even when $V$ itself is embedded in a larger graph $G$. In this paper we study decomposition of complete graphs into copies of $H$, such that the complements of those $H$ also form a decomposition of $G$.

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