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 G of order n at least 9, we show the existence of a feedback vertex set of order at most 1 3 ( n + 2 ¿ + m e + k 4 + ) , where ¿ is the number of loops of G , m e is the number of multiple edges of G , and k 4 + is the number of submultigraphs of G that arise from K 4 by subdividing one edge. This bound is best possible and implies several known bounds.