  Anharmonic phonon-phonon scattering serves a critical role in heat conduction
in solids. Previous studies have identified many selection rules for possible
phonon-phonon scattering channels imposed by phonon energy and momentum
conservation conditions and crystal symmetry. However, the
crystal-symmetry-based selection rules have mostly been \textit{ad hoc} so far
in selected materials, and a general formalism that can summarize known
selection rules and lead to new ones in any given crystal is still lacking. In
this work, we apply a general formalism for symmetry-based scattering selection
rules based on the group theory to anharmonic phonon-phonon scatterings, which
can reproduce known selection rules and guide the discovery of new selection
rules between phonon branches imposed by the crystal symmetry. We apply this
formalism to analyze the phonon-phonon scattering selection rules imposed by
the in-plane symmetry of graphene, and demonstrate the significant impact of
symmetry-breaking strain on the lattice thermal conductivity. Our work
quantifies the critical influence of the crystal symmetry on the lattice
thermal conductivity in solids and suggests routes to engineer heat conduction
by tuning the crystal symmetry.
