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The incipient infinite cluster appearing at the bond percolation thresholdcan be decomposed into singly-connected ``links'' and multiply-connected``blobs.'' Here we decompose blobs into objects known in graph theory as3-blocks. A 3-block is a graph that cannot be separated into disconnectedsubgraphs by cutting the graph at 2 or fewer vertices. Clusters, blobs, and3-blocks are special cases of $k$-blocks with $k=1$, 2, and 3, respectively. Westudy bond percolation clusters at the percolation threshold on 2-dimensionalsquare lattices and 3-dimensional cubic lattices and, using Monte-Carlosimulations, determine the distribution of the sizes of the 3-blocks into whichthe 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 aresignificantly smaller than the fractal dimensions of the blobs, making possiblemore efficient calculation of percolation properties. Additionally, thecloseness of the estimated values for $d_3$ in 2D and 3D is consistent with thepossibility that $d_3$ is dimension independent. Generalizing the concept ofthe backbone, we introduce the concept of a ``$k$-bone'', which is the set ofall points in a percolation system connected to $k$ disjoint terminal points(or sets of disjoint terminal points) by $k$ disjoint paths. We argue that thefractal dimension of a $k$-bone is equal to the fractal dimension of$k$-blocks, allowing us to discuss the relation between the fractal dimensionof $k$-blocks and recent work on path crossing probabilities.Comment: All but first 2 figs. are low resolution and are best viewed when  printe

Beyond blobs in percolation cluster structure: The distribution of 3-blocks at the percolation threshold