  The famous Gallai's Conjecture states that any connected graph with n
vertices has a path decomposition containing at most (n+1)/2 paths. In this
note, we explore graphs generated from removing edges from complete graphs. We
first provide an explicit construction for a path decomposition of complete
graphs that satisfies Gallai's Conjecture. We then use that construction to
prove that we can remove stars and certain tadpoles such that the resulting
graph still satisfies Gallai's Conjecture. We also introduce a potential
general approach through analyzing non-isomorphic path decompositions of
complete graphs.
