Feedback vertex sets in cubic multigraphs

A set of vertices of a multigraph whose removal produces a forest is a feedback vertex set. For a connected cubic multigraph GG of order nn at least 9, we show the existence of a feedback vertex set of order at most 13(n+2ℓ+me+k4+), where ℓℓ is the number of loops of GG, meme is the number of multiple edges of GG, and k4+ is the number of submultigraphs of GG that arise from K4K4 by subdividing one edge. This bound is best possible and implies several known bounds.