Silica (SiO2) is a versatile material with many different bulk polymorphs. Whereas bulk silica has been studied extensively, much less is known about silica clusters. The authors have studied silica clusters before, using global optimisation and a simple potential that had been reparameterised to get closer agreement with Density Functional Theory calculations [1,2]. This presentation focuses on a study of a special class of cluster geometries: fully-coordinated silica clusters, i.e. geometries where each silicon atom is chemically bonded to four oxygen atoms and each oxygen atom is bonded to two silicon atoms. While this is a common feature in the bulk, defects such as dangling oxygens are often found in clusters. In this presentation we will be discussing an algorithm for specifically generating fully-coordinated cluster geometries. In our earlier study of silica clusters we used the standard Basin Hopping algorithm, which is based on performing Monte Carlo moves in coordinate-space followed by local optimisations. In our present study we propose to perform Monte Carlo sampling on the set of graphs (i.e. the networks of silicon-oxygen bonds) rather than in coordinate-space, since it is much easier to design Monte Carlo moves in graph-space that retain fully-coordinatedness than in coordinate-space. At each step a three dimensional realisation of the graph is sought, using a cascade of optimisations involving two cost-functions and our silica potential. With this method we have generated databases of low-energy fully-coordinated cluster geometries in a size range of up to 30 SiO2 units. The proposed method can be generalised to other network-forming systems.