Let $G$ be a finite simple graph and $I(G)$ be the edge ideal of $G$. Suppose we can write $G=H\sqcup K$ as a disjoint union of subgraphs. If $\beta_{i,j}$ is the graded betti numbers then is it true that $$\beta_{i,j}(I(G))=\beta_{i,j}(I(H))+\beta_{i,j}(I(K))?$$
It will be also helpful if you can suggest me some survey article for betti numbers of edge ideals of graphs.
Thank you in advance.