eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-08-18
54:1
54:13
10.4230/LIPIcs.MFCS.2021.54
article
Perfect Forests in Graphs and Their Extensions
Gutin, Gregory
1
Yeo, Anders
2
3
https://orcid.org/0000-0003-0293-8708
Royal Holloway, University of London, UK
University of Southern Denmark, Odense, Denmark
University of Johannesburg, South Africa
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol202-mfcs2021/LIPIcs.MFCS.2021.54/LIPIcs.MFCS.2021.54.pdf
graphs
odd degree subgraphs
perfect forests
polynomial algorithms