  The nature of the coupling leading to superconductivity in layered materials
such as high-Tc superconductors and graphite intercalation compounds (GICs) is
still unresolved. In both systems, interactions of electrons with either
phonons or other electrons or both have been proposed to explain
superconductivity. In the high-Tc cuprates, the presence of a Van Hove
singularity (VHS) in the density of states near the Fermi level was long ago
proposed to enhance the many-body couplings and therefore may play a role in
superconductivity. Such a singularity can cause an anisotropic variation in the
coupling strength, which may partially explain the so-called nodal-antinodal
dichotomy in the cuprates. Here we show that the topology of the graphene band
structure at dopings comparable to the GICs is quite similar to that of the
cuprates and that the quasiparticle dynamics in graphene have a similar
dichotomy. Namely, the electron-phonon coupling is highly anisotropic,
diverging near a saddle point in the graphene electronic band structure. These
results support the important role of the VHS in layered materials and the
possible optimization of Tc by tuning the VHS with respect to the Fermi level.
