Large Induced Acyclic and Outerplanar Subgraphs of 2-Outerplanar Graph

Abstract

Albertson and Berman conjectured that every planar graph has an induced forest on half of its vertices. The best known lower bound, due to Borodin, is that every planar graph has an induced forest on two fifths of its vertices. In a related result, Chartran and Kronk, proved that the vertices of every planar graph can be partitioned into three sets, each of which induce a forest. We show tighter results for 2-outerplanar graphs. We show that every 2-outerplanar graph has an induced forest on at least half the vertices by showing that its vertices can be partitioned into two sets, each of which induces a forest. We also show that every 2-outerplanar graph has an induced outerplanar graph on at least two-thirds of its vertices.

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Notes

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    Formal definitions of graph theoretic terms will be given at the end of this section.

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Acknowledgements

The authors thank conversations with Bigong Zheng and Kai Lei. We thank anonymous reviewers for comments that help improving the presentation of this paper.

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Correspondence to Hung Le.

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This material is based upon work supported by the National Science Foundation under Grant Nos. CCF-1252833 and DMS-1359173. Melissa Sherman-Bennett performed this work while participating in the Summer 2015 REU program in Mathematics and Computer Science at Oregon State University.

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Borradaile, G., Le, H. & Sherman-Bennett, M. Large Induced Acyclic and Outerplanar Subgraphs of 2-Outerplanar Graph. Graphs and Combinatorics 33, 1621–1634 (2017). https://doi.org/10.1007/s00373-017-1859-3

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Keywords

  • Induced forest
  • Planar graphs
  • Outerplanarity
  • Albertson-Berman conjecture