A better approximation algorithm for finding planar subgraphs
Journal of Algorithms
An Algorithm for Finding Large Induced Planar Subgraphs
GD '01 Revised Papers from the 9th International Symposium on Graph Drawing
Splitting Number is NP-complete
WG '98 Proceedings of the 24th International Workshop on Graph-Theoretic Concepts in Computer Science
Node-and edge-deletion NP-complete problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
Disjoint Triangles of a Cubic Line Graph
Graphs and Combinatorics
On the complexity of the approximation of nonplanarity parameters for cubic graphs
Discrete Applied Mathematics - Brazilian symposium on graphs, algorithms and combinatorics
Acyclic colorings of subcubic graphs
Information Processing Letters
On maximum planar induced subgraphs
Discrete Applied Mathematics - Special issue: Traces of the Latin American conference on combinatorics, graphs and applications: a selection of papers from LACGA 2004, Santiago, Chile
Random Structures & Algorithms
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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 6/5- and 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.