  We study homomorphism problems of signed graphs. A signed graph is an
undirected graph where each edge is given a sign, positive or negative. An
important concept for signed graphs is the operation of switching at a vertex,
which is to change the sign of each incident edge. A homomorphism of a graph is
a vertex-mapping that preserves the adjacencies; in the case of signed graphs,
we also preserve the edge-signs. Special homomorphisms of signed graphs, called
s-homomorphisms, have been studied. In an s-homomorphism, we allow, before the
mapping, to perform any number of switchings on the source signed graph. This
concept has been extensively studied, and a full complexity classification
(polynomial or NP-complete) for s-homomorphism to a fixed target signed graph
has recently been obtained. Such a dichotomy is not known when we restrict the
input graph to be planar (not even for non-signed graph homomorphisms).
  We show that deciding whether a (non-signed) planar graph admits a
homomorphism to the square $C_t^2$ of a cycle with $t\ge 6$, or to the circular
clique $K_{4t/(2t-1)}$ with $t\ge2$, are NP-complete problems. We use these
results to show that deciding whether a planar signed graph admits an
s-homomorphism to an unbalanced even cycle is NP-complete. (A cycle is
unbalanced if it has an odd number of negative edges). We deduce a complete
complexity dichotomy for the planar s-homomorphism problem with any signed
cycle as a target.
  We also study further restrictions involving the maximum degree and the girth
of the input signed graph. We prove that planar s-homomorphism problems to
signed cycles remain NP-complete even for inputs of maximum degree~$3$ (except
for the case of unbalanced $4$-cycles, for which we show this for maximum
degree~$4$). We also show that for a given integer $g$, the problem for signed
bipartite planar inputs of girth $g$ is either trivial or NP-complete.
