  We use the deuteration of $c$-$\mathrm{C_{3}H_{2}}$ to probe the physical
parameters of starless and protostellar cores, related to their evolutionary
states, and compare it to the $\mathrm{N_{2}H^{+}}$-deuteration in order to
study possible differences between the deuteration of C- and N-bearing species.
We observed the main species $c$-$\mathrm{C_{3}H_{2}}$, the singly and doubly
deuterated species $c$-$\mathrm{C_{3}HD}$ and $c$-$\mathrm{C_{3}D_{2}}$, as
well as the isotopologue $c$-$\mathrm{{H^{13}CC_{2}H}}$ toward 10 starless
cores and 5 protostars in the Taurus and Perseus Complexes. We examined the
correlation between the
$N$($c$-$\mathrm{C_{3}HD}$)/$N$($c$-$\mathrm{C_{3}H_{2}}$) ratio and the dust
temperature along with the $\mathrm{H_2}$ column density and the CO depletion
factor. The resulting
$N$($c$-$\mathrm{C_{3}HD}$)/$N$($c$-$\mathrm{C_{3}H_{2}}$) ratio is within the
error bars consistent with $10\%$ in all starless cores with detected
$c$-$\mathrm{C_{3}HD}$. This also accounts for the protostars except for the
source HH211, where we measure a high deuteration level of $23\%$. The
deuteration of $\mathrm{N_{2}H^{+}}$ follows the same trend but is considerably
higher in the dynamically evolved core L1544. Toward the protostellar cores the
coolest objects show the largest deuterium fraction in
$c$-$\mathrm{C_{3}H_{2}}$. We show that the deuteration of
$c$-$\mathrm{C_{3}H_{2}}$ can trace the early phases of star formation and is
comparable to that of $\mathrm{N_{2}H^{+}}$. However, the largest
$c$-$\mathrm{C_{3}H_{2}}$ deuteration level is found toward protostellar cores,
suggesting that while $c$-$\mathrm{C_{3}H_{2}}$ is mainly frozen onto dust
grains in the central regions of starless cores, active deuteration is taking
place on ice.
