  A simple graph G=(V,E) is 3-rigid if its generic bar-joint frameworks in R3
are infinitesimally rigid. Block and hole graphs are derived from triangulated
spheres by the removal of edges and the addition of minimally rigid subgraphs,
known as blocks, in some of the resulting holes. Combinatorial
characterisations of minimal $3$-rigidity are obtained for these graphs in the
case of a single block and finitely many holes or a single hole and finitely
many blocks. These results confirm a conjecture of Whiteley from 1988 and
special cases of a stronger conjecture of Finbow-Singh and Whiteley from 2013.
