Counterexamples to the Albertson-Berman conjecture: minimum order, connectivity and an improved ratio bound

Abstract: In 1979, Albertson and Berman conjectured that every planar graph $G$ contains an induced forest of order at least $|V(G)|/2$. This long-standing conjecture was recently disproved by several explicit counterexamples, which naturally led to several extremal and structural questions that we answer. We combine mathematical arguments and exhaustive computations to show that the minimum order of a counterexample is $29$. We also construct infinitely many $4$-connected $5$-edge-connected counterexamples (and show that the unique such counterexample of minimum order has order $41$), whereas previously all known counterexamples had vertex-connectivity at most $3$. Furthermore, we construct an infinite family of planar graphs on $n$ vertices whose maximum induced forests have order at most $\frac{25}{52}n$, thereby improving the previous best upper bound. This family also yields infinitely many counterexamples (for every integer $d \geq 7$) to a conjecture of Chappell and Pelsmajer concerning induced forests of maximum degree at most $d$.