  Extending the notion of maxcut, the study of the frustration index of signed
graphs is one of the basic questions in the theory of signed graphs. Recently
two of the authors initiated the study of critically frustrated signed graphs.
That is a signed graph whose frustration index decreases with the removal of
any edge. The main focus of this study is on critical signed graphs which are
not edge-disjoint unions of critically frustrated signed graphs (namely
non-decomposable signed graphs) and which are not built from other critically
frustrated signed graphs by subdivision. We conjecture that for any given $k$
there are only finitely many critically $k$-frustrated signed graphs of this
kind.
  Providing support for this conjecture we show that there are only two of such
critically $3$-frustrated signed graphs where there is no pair of edge-disjoint
negative cycles. Similarly, we show that there are exactly ten critically
$3$-frustrated signed planar graphs that are neither decomposable nor
subdivisions of other critically frustrated signed graphs. We present a method
for building non-decomposable critically frustrated signed graphs based on two
given such signed graphs. We also show that the condition of being
non-decomposable is necessary for our conjecture.
