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) n2 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) n4 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) 25n-k10 and that the Albertson-Berman conjecture for simple graphs, assuming that it holds, would imply the lower bound a (M) n-k2 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) 310n+730 and give a construction of an infinite sequence of multigraphs with a (M) =37n+47.
Mikhail Makarov (Fri,) studied this question.