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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