Primitive star decompositions of graphs

A decomposition \(\mathcal{C}\) of a graph \(G\) is primitive if no proper, nontrivial subset of \(\mathcal{C}\) is a decomposition of an induced subgraph of \(G\). The existence of primitive decompositions has been studied for several decompositions, including path and cycle decompositions for complete and cocktail party graphs. In this work, we classify the existence of primitive star decompositions for complete graphs.