  Based on first-principles calculations and symmetry analysis, we propose that
a transition metal rutile oxide, in particular $\beta'$-PtO$_2$, can host a
three-dimensional topological Dirac semimetal phase. We find that
$\beta'$-PtO$_2$ possesses a linked nodal rings structure when spin-orbit
coupling is neglected. Incorporating spin-orbit coupling gaps the nodal rings,
while preserving a single pair of three-dimensional Dirac points protected by a
screw rotation symmetry. This Dirac point is created by a band inversion of two
$d$ bands, which is a realization of a DSM phase in correlated electron
systems. Moreover, a mirror plane in the momentum space carries a nontrivial
mirror Chern number $n_M = -2$, which distinguishes $\beta'$-PtO$_2$ from the
Dirac semimetals known so far, such as Na$_3$Bi and Cd$_3$As$_2$. If we apply a
perturbation that breaks the rotation symmetry and preserves the mirror
symmetry, the Dirac points are gapped and the system becomes a topological
crystalline insulator.
