  Let $G$ be a simple graph on the vertex set $\{v_{1},\ldots,v_{n}\}$. An
algebraic object attached to $G$ is the toric ideal $I_G$. We say that $I_G$ is
subgraph splittable if there exist subgraphs $G_1$ and $G_2$ of $G$ such that
$I_G=I_{G_1}+I_{G_2}$, where both $I_{G_1}$ and $I_{G_2}$ are not equal to
$I_G$. We show that $I_G$ is subgraph splittable if and only if it is edge
splittable. We also prove that the toric ideal of a complete bipartite graph is
not subgraph splittable. In contrast, we show that the toric ideal of a
complete graph $K_n$ is always subgraph splittable when $n \geq 4$.
Additionally, we show that the toric ideal of $K_n$ has a minimal splitting if
and only if $4 \leq n \leq 5$. Finally, we prove that any minimal splitting of
$I_G$ is also a reduced splitting.
