Abstract
A graph G is k-degenerated if it can be deleted by subsequent removals of vertices of degree k or less. We survey known results on the size of maximal k- degenerated induced subgraph in a planar graph. In addition, we sketch the proof that every planar graph of order n has a 4-degenerated induced subgraph of order at least 8/9 · n. We also show that in every planar graph with at least 7 vertices, deleting a suitable vertex allows us to subsequently remove at least 6 more vertices of degree four or less.
The first author acknowledges partial support from the research grant 7/TU/13 and from the APW grant ESF-EC-0009-10 within the EUROCORES Programme EUROGIGA (project GReGAS) of the European Science Foundation.
The work of the second author leading to this invention has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 259385.
The third author acknowledges partial support from the research grants NSF No. 11171730 and ZJNSF No. Z6110786.
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© 2013 Scuola Normale Superiore Pisa
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Lukot’ka, R., Mazák, J., Zhu, X. (2013). Degenerated induced subgraphs of planar graphs. In: Nešetřil, J., Pellegrini, M. (eds) The Seventh European Conference on Combinatorics, Graph Theory and Applications. CRM Series, vol 16. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-475-5_42
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DOI: https://doi.org/10.1007/978-88-7642-475-5_42
Publisher Name: Edizioni della Normale, Pisa
Print ISBN: 978-88-7642-474-8
Online ISBN: 978-88-7642-475-5
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