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

Abstract

A signed graph $(G,σ)$ is a graph $G$ with a signature $σ$ 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 $(G, σ)$ is the minimum number of negative edges among all signatures equivalent to $σ$. 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 $5$.

Figures (8)

• Figure 1: Signed graphs in $\mathcal{S}^*(3)$.
• Figure 2: Two types of bridge faces.
• Figure 3:
• Figure 4:
• Figure 5:

Theorems & Definitions (27)

• Definition 1.1
• Theorem 1.2
• Definition 1.5
• Conjecture 1.7
• Theorem 1.8
• Theorem 2.1
• proof
• proof
• Lemma 2.4
• Proposition 2.5