Satisfiability (SAT) solvers based on techniques such as conflict driven clause learning (CDCL) have produced excellent performance on both synthetic and real world industrial problems. While these CDCL solvers only operate on a per-problem basis, graph neural network (GNN) based solvers bring new benefits to the field by allowing practitioners to exploit knowledge gained from previously solved problems to expedite solving of new SAT problems. Prior works in the GNN for SAT space often improve the performance of GNN based solvers with novel architectures or large synthetic datasets. However, one specific area that is often studied in the context of CDCL solvers, but largely overlooked in GNN solvers, is the relationship between graph theoretic measure of structure in SAT problems and the generalisation ability of GNN solvers. To bridge the gap between structural graph properties (e.g., modularity, self-similarity) and the generalisability (or lack thereof) of GNN based SAT solvers, we present StructureSAT: a curated dataset, along with code to further generate novel examples, containing a diverse set of SAT problems from well known problem domains. Furthermore, we also utilise a novel splitting method that focuses on deconstructing the families into more detailed hierarchies based on their structural properties. With the new dataset, we aim to help explain problematic generalisation in existing GNN SAT solvers, and demonstrate an alternative approach to expedite GNN training efficiency by exploiting knowledge of structural graph properties. We conclude with multiple future directions that can help researchers in GNN based SAT solving develop more effective and generalisable SAT solvers.