  A \emph{signed graph} $(G, \sigma)$ is a graph $G$ together with an
assignment $\sigma:E(G) \rightarrow \{+,-\}$. The notion of homomorphisms of
signed graphs is a relatively new development which allows to strengthen the
connection between the theories of minors and colorings of graphs. Following
this thread of thoughts, we investigate this connection through the notion of
Extended Double Covers of signed graphs, which was recently introduced by
Naserasr, Sopena and Zaslavsky. More precisely, we say that a signed graph $(B,
\pi)$ is planar-complete if any planar signed graph $(G, \sigma)$ which
verifies the conditions of a basic no-homomorphism lemma with respect to
$(B,\pi)$ admits a homomorphism to $(B, \pi)$. Our conjecture then is that: if
$(B, \pi)$ is a connected signed graph with no positive odd closed walk which
is planar-complete, then its Extended Double Cover ${\rm EDC}(B,\pi)$ is also
planar-complete. We observe that this conjecture largely extends the Four-Color
Theorem and is strongly connected to a number of conjectures in extension of
this famous theorem.
  A given (signed) graph $(B,\pi)$ \emph{bounds} a class of (signed) graphs if
every (signed) graph in the class admits a homomorphism to $(B,\pi)$. In this
work, and in support of our conjecture, we prove it for the subclass of signed
$K_4$-minor free graphs. Inspired by this development, we then investigate the
problem of finding optimal homomorphism bounds for subclasses of signed
$K_4$-minor-free graphs with restrictions on their girth and we present nearly
optimal solutions. Our work furthermore leads to the development of weighted
signed graphs.
