  Given a finitely generated relatively hyperbolic group $G$, we construct a
finite generating set $X$ of $G$ such that $(G,X)$ has the `falsification by
fellow traveler property' provided that the parabolic subgroups
$\{H_\omega\}_{\omega\in \Omega}$ have this property with respect to the
generating sets $\{X\cap H_\omega\}_{\omega\in \Omega}$. This implies that
groups hyperbolic relative to virtually abelian subgroups, which include all
limit groups and groups acting freely on $\mathbb{R}^n$-trees, or geometrically
finite hyperbolic groups, have generating sets for which the language of
geodesics is regular, and the complete growth series and complete geodesic
series are rational. As an application of our techniques, we prove that if each
$H_\omega$ admits a geodesic biautomatic structure over $X\cap H_\omega$, then
$G$ has a geodesic biautomatic structure.
  Similarly, we construct a finite generating set $X$ of $G$ such that $(G,X)$
has the `bounded conjugacy diagrams' property or the `neighbouring shorter
conjugate' property if the parabolic subgroups $\{H_\omega\}_{\omega\in
\Omega}$ have this property with respect to the generating sets $\{X\cap
H_\omega\}_{\omega\in \Omega}$. This implies that a group hyperbolic relative
to abelian subgroups has a generating set for which its Cayley graph has
bounded conjugacy diagrams, a fact we use to give a cubic time algorithm to
solve the conjugacy problem.
  Another corollary of our results is that groups hyperbolic relative to
virtually abelian subgroups have a regular language of conjugacy geodesics.
