License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2021.54
URN: urn:nbn:de:0030-drops-144947
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14494/
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Gutin, Gregory ; Yeo, Anders

Perfect Forests in Graphs and Their Extensions

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LIPIcs-MFCS-2021-54.pdf (0.7 MB)

Abstract

Let G be a graph on n vertices. For i ∈ {0,1} and a connected graph G, a spanning forest F of G is called an i-perfect forest if every tree in F is an induced subgraph of G and exactly i vertices of F have even degree (including zero). An i-perfect forest of G is proper if it has no vertices of degree zero. Scott (2001) showed that every connected graph with even number of vertices contains a (proper) 0-perfect forest. We prove that one can find a 0-perfect forest with minimum number of edges in polynomial time, but it is NP-hard to obtain a 0-perfect forest with maximum number of edges. We also prove that for a prescribed edge e of G, it is NP-hard to obtain a 0-perfect forest containing e, but we can find a 0-perfect forest not containing e in polynomial time. It is easy to see that every graph with odd number of vertices has a 1-perfect forest. It is not the case for proper 1-perfect forests. We give a characterization of when a connected graph has a proper 1-perfect forest.

BibTeX - Entry

@InProceedings{gutin_et_al:LIPIcs.MFCS.2021.54,
  author =	{Gutin, Gregory and Yeo, Anders},
  title =	{{Perfect Forests in Graphs and Their Extensions}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{54:1--54:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14494},
  URN =		{urn:nbn:de:0030-drops-144947},
  doi =		{10.4230/LIPIcs.MFCS.2021.54},
  annote =	{Keywords: graphs, odd degree subgraphs, perfect forests, polynomial algorithms}
}

Keywords: graphs, odd degree subgraphs, perfect forests, polynomial algorithms
Collection: 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)
Issue Date: 2021
Date of publication: 18.08.2021

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