Fig. 9

Patterns free of triplets and symmetric squares showing a \(\check{\mathbb {D}}\)-edge uv in two distinct intersecting \(\{4,\!6\}\)-cycles which themselves do not intersect 2-cycles. (i–iii) The edge uv connects two distinct squares and is part of two \(\{4,\!6\}\)-cycles whose intersection is only uv. (iv) The edge uv is part of two 6-cycles whose intersection is a \(\check{\mathbb {D}}\mathbb {S}\check{\mathbb {D}}\)-path starting in uv. Here one square (marked in blue) is clearly fixed: if this square could be switched, this would merge each of the two existing 6-cycles into a longer cycle