Large induced forests in planar multigraphs

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 $G$ on $n$ vertices, denote by $a(G)$ the number of vertices in the largest induced forest in $G$. The Albertson-Berman conjecture, which has been open since 1979, states that $a(G) \geq \frac{n}{2}$ for every simple planar graph $G$. 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 $a(M) \geq \frac{n}{4}$ for every planar multigraph $M$ 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 $k$, is small. In particular, we prove the lower bound $a(M) \geq \frac{2}{5}n-\frac{k}{10}$ and that the Albertson-Berman conjecture for simple graphs, assuming that it holds, would imply the lower bound $a(M) \geq \frac{n-k}{2}$ for multigraphs, which would be better than the general lower bound when $k$ is small. Finally, we study the variant of the problem where the plane multigraphs are prohibited from having $2$-faces, which is the main non-trivial problem that we introduce in this article. For that variant without $2$-faces, we prove the lower bound $a(M) \geq \frac{3}{10}n+\frac{7}{30}$ and give a construction of an infinite sequence of multigraphs with $a(M)=\frac{3}{7}n+\frac{4}{7}$.