Frustration-critical signed graphs

A signed graph (G,Σ) is a G together with set Σ⊆E(G) of negative edges. circuit positive if the product signs its edges positive. balanced all circuits are The frustration index l(G,Σ) minimum cardinality E⊆E(G) such that (G−E,Σ−E) balanced, and k-critical l(G,Σ)=k l(G−e,Σ−e)<k, for every e∈E(G). We study decomposition subdivision critical graphs completely determine t-critical graphs, t≤2. Critical characterized. then focus on non-decomposable graphs. In particular, we characterize S∗ not containing decomposable subgraph t≤k. prove consists cyclically 4-edge-connected projective-planar cubic Furthermore, construct k≥1.