  Denote by $\lambda K_v$ the complete graph of order $v$ with multiplicity
$\lambda$. Let $\lambda K_v-\lambda K_w-\lambda K_u$ be the graph obtained from
$\lambda K_v$ by the removal of the edges of two vertex disjoint complete
multi-subgraphs with multiplicity $ \lambda $ of orders $ w $ and $ u $,
respectively. When $\lambda$ is odd, it is shown that there exists a triangle
decomposition of $\lambda K_v-\lambda K_w-\lambda K_u$ if and only if $v\geq
w+u+\max\{u,w\}$, $ \lambda \left({v\choose 2}-{u\choose 2}-{w\choose 2}\right)
\equiv 0 \pmod 3$ and $\lambda (v-w) \equiv \lambda (v-u) \equiv \lambda (v-1)
\equiv 0 \pmod 2$. When $\lambda$ is even, it is shown that for large enough
$v$, the elementary necessary conditions for the existence of a triangle
decomposition of $\lambda K_v-\lambda K_w-\lambda K_u$ are also sufficient.
