  The incipient infinite cluster appearing at the bond percolation threshold
can be decomposed into singly-connected ``links'' and multiply-connected
``blobs.'' Here we decompose blobs into objects known in graph theory as
3-blocks. A 3-block is a graph that cannot be separated into disconnected
subgraphs by cutting the graph at 2 or fewer vertices. Clusters, blobs, and
3-blocks are special cases of $k$-blocks with $k=1$, 2, and 3, respectively. We
study bond percolation clusters at the percolation threshold on 2-dimensional
square lattices and 3-dimensional cubic lattices and, using Monte-Carlo
simulations, determine the distribution of the sizes of the 3-blocks into which
the blobs are decomposed. We find that the 3-blocks have fractal dimension
$d_3=1.2\pm 0.1$ in 2D and $1.15\pm 0.1$ in 3D. These fractal dimensions are
significantly smaller than the fractal dimensions of the blobs, making possible
more efficient calculation of percolation properties. Additionally, the
closeness of the estimated values for $d_3$ in 2D and 3D is consistent with the
possibility that $d_3$ is dimension independent. Generalizing the concept of
the backbone, we introduce the concept of a ``$k$-bone'', which is the set of
all points in a percolation system connected to $k$ disjoint terminal points
(or sets of disjoint terminal points) by $k$ disjoint paths. We argue that the
fractal dimension of a $k$-bone is equal to the fractal dimension of
$k$-blocks, allowing us to discuss the relation between the fractal dimension
of $k$-blocks and recent work on path crossing probabilities.
