Large induced forests in planar multigraphs
Mikhail Makarov

TL;DR
This paper investigates lower bounds on the size of the largest induced forests in planar multigraphs, relating them to the Albertson-Berman conjecture and exploring special cases and variants.
Contribution
It establishes new lower bounds for planar multigraphs, connects these bounds to the conjecture for simple graphs, and examines multigraphs without 2-faces.
Findings
Proves that $a(M) \,\geq\, \frac{n}{4}$ for all planar multigraphs, and this bound is tight.
Derives bounds depending on the number of parallel edges, such as $a(M) \,\geq\, \frac{2}{5}n - \frac{k}{10}$.
Analyzes multigraphs without 2-faces, proving $a(M) \,\geq\, \frac{3}{10}n + \frac{7}{30}$ and constructing examples with $a(M) = \frac{3}{7}n + \frac{4}{7}$.
Abstract
For a graph on vertices, denote by the number of vertices in the largest induced forest in . The Albertson-Berman conjecture, which has been open since 1979, states that for every simple planar graph . We show that the version of this problem for multigraphs (allowing parallel edges) is easily reduced to the problem about the independence number of simple planar graphs. Specifically, we prove that for every planar multigraph and that this lower bound is tight. Then, we study the case when the number of pairs of vertices with parallel edges, which we denote by , is small. In particular, we prove the lower bound and that the Albertson-Berman conjecture for simple graphs, assuming that it holds, would imply the lower bound for multigraphs, which would…
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