Finiteness of non-decomposable critically 4 and 5-frustrated signed graphs

A signed graph $(G,\sigma)$ is a graph $G$ with a signature $\sigma$ labeling each edge with a positive or negative sign. Two signatures of $G$ are switching equivalent if one is obtained from the other by changing the signs of all edges in an edge-cut. The frustration index of a signed graph $(G, \sigma)$ is the minimum number of negative edges among all signatures equivalent to $\sigma$. A signed graph is critically $k$-frustrated if it has frustration index $k$, and the removal of any edge decreases its frustration index. A critically $k$-frustrated signed graph is prime if it has no subdivided edge (including multiedge) and none of its subgraphs is the edge-disjoint union of critically frustrated signed graphs. Steffen and Naserasr et al. conjectured that for any positive integer $k$, there are finitely many prime critically $k$-frustrated signed graphs. The cases $k=1,2,3$ have been proved to be true recently by Cappello et al.. In this paper, we show that the conjecture holds when $k=4$ and $5$.