  We present a Monte Carlo study of the backbone and the shortest-path
exponents of the two-dimensional $Q$-state Potts model in the Fortuin-Kasteleyn
bond representation. We first use cluster algorithms to simulate the critical
Potts model on the square lattice and obtain the backbone exponents $d_{\rm B}
= 1.732 \, 0(3)$ and $1.794(2)$ for $Q=2,3$ respectively. However, for large
$Q$, the study suffers from serious critical slowing down and slowly converging
finite-size corrections. To overcome these difficulties, we consider the O$(n)$
loop model on the honeycomb lattice in the densely packed phase, which is
regarded to correspond to the critical Potts model with $Q=n^2$. With a highly
efficient cluster algorithm, we determine from domains enclosed by the loops
$d_{\rm B} =1.643\,39(5), 1.732\,27(8), 1.793\,8(3), 1.838\,4(5), 1.875\,3(6)$
for $Q = 1, 2, 3, 2 \! + \! \sqrt{3}, 4$, respectively, and $d_{\rm min} =
1.094\,5(2), 1.067\,5(3), 1.047\,5(3), 1.032\,2(4)$ for $Q=2,3, 2+\sqrt{3}, 4$
respectively. Our estimates significantly improve over the existing results for
both $d_{\rm B}$ and $d_{\rm min}$. Finally, by studying finite-size
corrections in backbone-related quantities, we conjecture an exact formula as a
function of $n$ for the leading correction exponent.
