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/
Gutin, Gregory ;
Yeo, Anders
Perfect Forests in Graphs and Their Extensions
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 |