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# Perfect Forests in Graphs and Their Extensions

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LIPIcs.MFCS.2021.54.pdf
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## Cite As

Gregory Gutin and Anders Yeo. Perfect Forests in Graphs and Their Extensions. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 54:1-54:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.MFCS.2021.54

## 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.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graph theory
##### Keywords
• graphs
• odd degree subgraphs
• perfect forests
• polynomial algorithms

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## References

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