A 15/31 Family of Maximal Planar Graphs Disproving the Albertson–Berman Conjecture

We disprove the Albertson–Berman conjecture (1979), which asserts that every n-vertex planar graph has an induced forest on at least n/2 vertices. For every k ≥ 2, we construct a simple 3-connected maximal planar graph Mₖ on 31k vertices whose maximum induced forest has exactly 15k vertices, giving the asymptotic ratio 15/31 < 1/2. The smallest member is a 31-vertex plane triangulation. A self-contained Python verifier is included.

Correspondence: hdhehia@knou.ac.kr