A counterexample to the Albertson-Berman conjecture about induced forests in planar graphs

For a graph G, denote by $a(G)$ the number of vertices in the largest induced forest in G. The Albertson-Berman conjecture, which had been open since 1979, states that a(G) ≥ n/2 for every simple planar graph G on n vertices. Although the Albertson-Berman conjecture was recently resolved in the negative by constructing a counterexample with the help of AI, we independently found a counterexample to the Albertson-Berman conjecture without AI and present it in this article. Our counterexample is on $39$ vertices with $a(G)=19$. Finally, we indicate, without a full proof, a variant of this construction with a larger number of vertices, but with a slightly smaller ratio a(G)/n=37/76.