In this paper, a series of advances in Elastic Graph Matching (EGM) for face recognition assisted by the availability of 3D facial geometry is proposed. More specifically, we conceptually extend the EGM algorithm in order to exploit the 3D nature of human facial geometry for face recognition/verification. In order to achieve that, first we extend the matching module of the EGM algorithm in order to capitalize on the 2.5D facial data. Moreover, we incorporate the 3D geometry into the multiscale analysis used and build a novel geodesic multiscale morphological pyramid of dilations/erosions in order to fill the graph jets. We demonstrate the efficiency of the proposed advances in the face recognition/verification problem.