Assume that a Heegaard diagram $(\Sigma_g,\{\alpha_1,\ldots,\alpha_g\},\{\beta_1,\ldots,\beta_g\})$, defining a Heegaard splitting of a closed 3-manifold, is given. So, by attaching 3-dimensional 2-handles to $\Sigma_g$ along $\alpha_i$ and $\beta_i$, one gets a closed 3-manifold with the Heegaard surface $\Sigma_g$.

I wonder if there are any answers to following questions:

Are there sufficient conditions on the diagram for the Heegaard splitting to be irreducible? Furthermore, is there an algorithm to detect whether the splitting is irreducible or not?

reducibleif there are disks $D_i\subset H_i$ such that $\partial D_1=\partial D_2$. (@mike) $\endgroup$ – Mustafa Jul 1 '17 at 19:33