Frustration index of a signed planar graph and the feedback vertex set

A feedback vertex set of a graph is a set of vertices whose deletion leaves a forest. In 2016, Dross, Montassier, and Pinlou conjectured that every planar graph $G$ of girth at least $g$ admits a feedback vertex set of size at most $e(G)/g$. In this note, we confirm this conjecture by connecting this problem with signed graphs. The frustration index of a signed graph $(G,\Sigma)$ is defined as the minimum number of negative edges among all signatures on $G$ that are switching-equivalent to $\Sigma$. Equivalently, it is the minimum number of edges whose deletion results in a balanced subgraph of $(G,\Sigma)$. We show that the minimum size of a feedback vertex set of a planar graph is bounded above by the maximum frustration index over all signatures of the graph, and thereby provide a tight upper bound on the size of the minimum feedback vertex set, which resolves the conjecture of Dross, Montassier, and Pinlou (2016).