Critical geometry of triconnected clusters in percolation

We investigate the critical geometry of triconnected clusters in percolation, which remain connected after the removal of any two sites. By combining biconnected-component decomposition with the SPQR decomposition of graphs, we develop an efficient algorithm for identifying triconnected clusters in large systems. Extensive Monte Carlo simulations yield estimates of the fractal dimension d₃=1.2308(3), $1.083(2)$, and $0.77(2)$ in dimensions $d=2$, $3$, and $4$, respectively, revealing a systematic decrease of d₃ with increasing d. In dimension $d=6$, the largest triconnected cluster grows logarithmically with system size, whereas on complete graphs, corresponding to d=∞, it approaches a finite size. Both behaviors indicate a vanishing fractal dimension, d₃=0, for d≥6. Furthermore, the singular contribution to the susceptibility-like quantity associated with the second moment of the triconnected-cluster size distribution scales as χ~ L2d₃-d. Consequently, it vanishes for d≥3, leaving a finite asymptotic susceptibility, in contrast to the divergent behavior observed in standard percolation.