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Every lattice for which the bond percolation critical probability can befound exactly possesses a critical polynomial, with the root in [0,1] providingthe threshold. Recent work has demonstrated that this polynomial may begeneralized through a definition that can be applied on any periodic lattice.The polynomial depends on the lattice and on its decomposition into identicalfinite subgraphs, but once these are specified, the polynomial is essentiallyunique. On lattices for which the exact percolation threshold is unknown, thepolynomials provide approximations for the critical probability with theestimates appearing to converge to the exact answer with increasing subgraphsize. In this paper, I show how this generalized critical polynomial can beviewed as a graph invariant, similar to the Tutte polynomial. In particular,the critical polynomial is computed on a finite graph and may be found usingthe recursive deletion-contraction algorithm. This allows calculation on acomputer, and I present such results for the kagome lattice using subgraphs ofup to 36 bonds. For one of these, I find the prediction p_c=0.52440572...,which differs from the numerical value, p_c=0.52440503(5), by only 6.9 x10^{-7}.

The percolation critical polynomial as a graph invariant