  We present a Monte Carlo study of the Fortuin-Kasteleyn (FK) clusters of the
Ising model on the square (2D) and simple-cubic (3D) lattices. The wrapping
probability, a dimensionless quantity characterizing the topology of the FK
clusters on a torus, is found to suffer from smaller finite-size corrections
than the well-known Binder ratio, and yields a high-precision critical coupling
as $K_c(3\rm D)=0.221\,654\,631(8)$. We then study geometric properties of the
FK clusters at criticality. It is demonstrated that the distribution of the
critical largest-cluster size $C_1$ follows a single-variable function as
$P(C_1,L){\rm d}C_1=\tilde P(x){\rm d}x$ with $x\equiv C_1/L^{d_{\rm F}}$ ($L$
is the linear size), and that the fractal dimension $d_{\rm F}$ is identical to
the magnetic exponent. An interesting bimodal feature is observed in
distribution $\tilde P(x)$ in 3D, and attributed to the different approaching
behaviors for $K \to K_c+0^\pm$. For a critical FK configuration, the cluster
number per site $n(s,L)$ of size $s$ is confirmed to obey the standard scaling
form $n(s,L)\sim s^{-\tau}\tilde n(s/L^{d_{\rm F}})$, with hyper-scaling
relation $\tau=1+d/d_{\rm F}$ and the spatial dimension $d$. To further
characterize the compactness of the FK clusters, we measure their graph
distances and determine the shortest-path exponents as $d_{\rm min}(3\rm
D)=1.259\,4(2)$ and $d_{\rm min}(2\rm D)=1.094\,0(3)$. Further, by excluding
all the bridges from the occupied bonds, we obtain bridge-free configurations
and determine the backbone exponents as $d_{\rm B}(3\rm D)=2.167\,3(15)$ and
$d_{\rm B}(2\rm D)=1.732\,1(4)$. The estimates of the universal wrapping
probabilities for the 3D Ising model and of the geometric critical exponents
$d_{\rm min}$ and $d_{\rm B}$ either improve over the existing results or have
not been reported yet.
