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 \(n \ge 4\) . Additionally, we show that the toric ideal of \(K_n\) has a minimal splitting if and only if \(4 \le n \le 5\) . Finally, we prove that any minimal splitting of \(I_G\) is also a reduced splitting.
