Planarization and Acyclic Colorings of Subcubic Claw-Free Graphs

We study methods of planarizing and acyclically coloring claw-free subcubic graphs. We give a polynomial-time algorithm that, given such a graph G, produces an independent set Q of at most n/6 vertices whose removal from G leaves an induced planar subgraph P (in fact, P has treewidth at most four). We further show the stronger result that in polynomial-time a set of at most n/6 edges can be identified whose removal leaves a planar subgraph (of treewidth at most four). From an approximability point of view, we show that our results imply 6/5and 9/8-approximation algorithms, respectively, for the (NP-hard) problems of finding a maximum induced planar subgraph and a maximum planar subgraph of a subcubic claw-free graph, respectively. Regarding acyclic colorings, we give a polynomial-time algorithm that finds an optimal acyclic vertex coloring of a subcubic claw-free graph. To our knowledge, this represents the largest known subclass of subcubic graphs such that an optimal acyclic vertex coloring can be found in polynomial-time. We show that this bound is tight by proving that the problem is NP-hard for cubic line graphs (and therefore, claw-free graphs) of maximum degree d ≥ 4. An interesting corollary to the algorithm that we present is that there are exactly three subcubic claw-free graphs that require four colors to be acyclically colored. For all other such graphs, three colors suffice.