The signed graph $(G,\Sigma)$ is the combination of the graph $G$ and the set of negative edges $\Sigma \subseteq E(G)$. The circuit is positive if the product of the signs of the edges is positive. A signed graph $(G,\Sigma)$ is balanced if all its circuits are positive. The frustration index $l(G,\Sigma)$ is the set $E \subseteq E(G)$ in which $(GE,\Sigma-E)$ is balanced and $(G,\Sigma)$ Minimum cardinality. $l(G,\Sigma) = k$ and $l(Ge, \Sigma – e)
We study the decomposition and subdivision of the significant signed graph, and fully determine the set of $t$-significant signed graphs for $t \leq 2$. Significant signed graphs are characterized. Next, we focus on non-decomposable significant signed graphs. In particular, we characterize the set $S^*$ of non-decomposable $k$-critical signed graphs that do not contain any decomposable $t$-critical signed subgraphs for each $t\leq k$. We prove that $S^*$ consists of a projective planar cubic graph with periodic 4-edge connections. Additionally, we create a $k$-critical signed graph of $S^*$ for every $k\geq 1$.
