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?



Irreducibility of a Heegaard splitting from its Heegaard diagram
There is a criterion which is usually used to show that a Heegaard splitting is irreducible, which in fact shows that the Heegaard splitting is strongly irreducible (meaning that any pair of meridian disks for $H\_1$ and $H\_2$ have intersecting boundaries). The original criterion was given by Casson and Gordon in an unpublished paper (see the appendix of [Schultens-Moriah](https://doi.org/10.1016/S0040-9383(97)00072-4)). 


The paper of [Lustig-Moriah](https://doi.org/10.1016/j.top.2004.01.004) discusses the Casson-Gordon "rectangle condition" further. 


Some other sufficient conditions are given by [Jung Hoon Lee](https://doi.org/10.1017/S0305004110000095). 


Finally, it is now known how to classify irreducible Heegaard splittings of non-Haken hyperbolic 3-manifolds by [Colding-Gabai-Ketover](https://arxiv.org/abs/1509.05945). They show how to give a list (without repetition) of Heegaard splittings in these 3-manifolds. The proof shows how to recognize whether a Heegaard splitting is irreducible, and hence answers your question in this special case. 


