  Let $\Gamma$ be a triangulation of a connected closed $2$-dimensional (not
necessarily orientable) surface. Using zigzags (closed left-right paths), for
every face of $\Gamma$ we define the $z$-monodromy which acts on the oriented
edges of this face. There are precisely $7$ types of $z$-monodromies. We
consider the following two cases: (M1) the $z$-monodromy is identity, (M2) the
$z$-monodromy is the consecutive passing of the oriented edges. Our main result
is the following: the subgraphs of the dual graph $\Gamma^{*}$ formed by edges
whose $z$-monodromies are of types (M1) and (M2), respectively, both are
forests. We apply this statement to the connected sum of $z$-knotted
triangulations.
