Albertson–Berman inequality for 4-connected planar graphs

Determine whether every 4-connected planar graph G satisfies the Albertson–Berman inequality a(G) ≥ |V(G)|/2.

Background

The paper disproves the Albertson–Berman conjecture for simple planar graphs by constructing graphs M_k with induced-forest ratio 15/31, below 1/2. However, the construction relies on separating triangles: in particular, the triangle P_1 separates the vertices of one seed from the remaining seeds.

Because the constructed counterexamples are not 4-connected, they do not resolve whether the original inequality might still hold for the more restricted class of 4-connected planar graphs. The authors leave this as an explicit unresolved question.