We show that f (x) = ⌊ 2 x⌋ is a θ-bounding function for the class of subcubic graphs and that it is best possible. This generalizes a result by Henning et al. (2012), who showed that θ (G) ≤ 2α(G) for any subcubic triangle-free graph G. Moreover, we provide a θ-bounding function for the class of K4-free graphs with maximum degree at most 4. Finally, we study the problem Clique Cover for subclasses of planar graphs and graphs with bounded maximum degree: in particular, answering a question of Cerioli et al. (2008), we show it admits a PTAS for planar graphs. © 2017 Elsevier B.V. All rights reserved.