Our paper is organized as follows: The main findings of I are summarized in Sec. II. We describe in Sec. III the algorithm we use to obtain the expressions of A, B, C for the Potts model with pure 2- and/or 3-site subnet interactions. The resulting expressions of A, B, C are listed in the Appendix. In Sec. IV, we describe the transfer matrix technique and the finite-size scaling method. Numerical results of our transfer matrix calculations and finite-size scaling analysis are given in Sec. V. New exact thresholds are also given in Sec. V for site percolation on kagome-type (1 × 1) : ( $^{n}$× n ) subnet lattices for n up to 6. We summarize our main findings in Sec. VI. where the products are taken over the up-pointing triangles. Wu [9, 12] showed that, in the regime in which the ground state of W$_{△}$ is ferromagnetic, the exact critical frontier is given by The critical function (4) yields the exact thresholds of site percolation on lattices gen- erated from triangular-type lattices. Consider a Potts model on a triangular subnet lattice with pure 3-site interactions in doted triangles shown in Fig. 2. Regarding faces of 3-spin interactions in an n × n subnet as sites of a new lattice, the Potts model maps to a site $^{percolation on a (1}$× 1) : ( $^{n}$− $^{1)}$× ( $^{n}$− 1) kagome-type subnet lattice. The critical frontier (4) then gives the exact threshold of the site percolation. Examples of the mapping are shown in Fig. 3 and 4 for n = 2 and 3. For kagome-type lattices shown in Fig. 1(b) the partition function is 4 Z$_{kag}$ ( q ; A, B, C ; A $^{′}$, B $^{′}$, C $^{′}$) = q ∑ τ i =1 [ ∏ △ W$_{△}$ ( i, j, k ) ] · [ ∏ ▽ W$_{▽}$ ( i $^{′}$, j $^{′}$, k$^{′}$) ] . (5) qA = C. (4) 2 B + C > 0 , 3 B + C > 0 , (3) Z$_{tri}$ ( q ; A, B, C ) = q ∑ τ i =1 ∏ △ W$_{△}$ ( i, j, k ) , (2) II. MAIN RESULTS OF I We summarize in this section the main results of I. For the triangular-type lattice shown in Fig. 1(a), the partition function is