  A path in a graph $G$ is called non-self-touching if two vertices are
neighbours in the path if and only if they are neighbours in the graph. We
investigate the existence of doubly infinite non-self-touching paths in
infinite plane graphs.
  The matching graph $G_*$ of an infinite plane graph $G$ is obtained by adding
all diagonals to all faces, and it plays an important role in the theory of
site percolation on $G$. The main result of this paper is a necessary and
sufficient condition on $G$ for the existence of a doubly infinite
non-self-touching path in $G_*$ that traverses some diagonal. This is a key
step in proving, for quasi-transitive $G$, that the critical points of site
percolation on $G$ and $G_*$ satisfy the strict inequality $p_c(G_*) < p_c(G)$,
and it complements the earlier result of Grimmett and Li (Percolation critical
points of matching lattice pairs, arXiv:2205.02734), proved by different
methods, concerning the case of transitive graphs. Furthermore it implies, for
quasi-transitive graphs, that $p_u(G) + p_c(G) \ge 1$, with equality if and
only if the graph $G_\Delta$, obtained from $G$ by emptying all separating
triangles, is a triangulation. Here, $p_u$ is the critical probability for the
existence of a unique infinite open cluster.
