Strong Heegaard diagrams and strong L-spaces

We study a class of 3-manifolds called strong L-spaces, which by definition admit a certain type of Heegaard diagram that is particularly simple from the perspective of Heegaard Floer homology. We provide evidence for the possibility that every strong L-space is the branched double cover of an alternating link in the three-sphere. For example, we establish this fact for a strong L-space admitting a strong Heegaard diagram of genus two via an explicit classification. We also show that there exist finitely many strong L-spaces with bounded order of first homology; for instance, through order eight, they are connected sums of lens spaces. The methods are topological and graph theoretic. We discuss many related results and questions.