The independence number in graphs of maximum degree three

We prove that a image-free graph image of order image, size image and maximum degree at most three has an independent set of cardinality at least image, where image counts the number of components of image whose blocks are each either isomorphic to one of four specific graphs or edges between two of these four specific graphs and image is the maximum number of vertex-disjoint triangles in image. Our result generalizes a bound due to Heckman and Thomas [C.C. Heckman, R. Thomas, A new proof of the independence ratio of triangle-free cubic graphs, Discrete Math. 233 (2001) 233–237].