Upper-embeddability and the decycling number of connected 4-regular graphs Upper-embeddability and the decycling number of connected 4-regular graphs

The decycling number of a graph G , denoted by ∇ ( G ) , is the smallest number of vertices whose removal results in an acyclic subgraph of G . A decycling set S of G with ∇ ( G ) vertices is said to be a ∇ -set. For any connected loopless 4-regular graph G on n vertices, it is shown that n + 1 3 ≤...

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Veröffentlicht in:Graphs and combinatorics Jg. 41; H. 6; S. 132
Hauptverfasser: Long, Shude, Cai, Junliang
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Tokyo Springer Japan 01.12.2025
Springer Nature B.V
Schlagworte:
Apexes
Combinatorics
Engineering Design
Graph theory
Graphs
Mathematics
Mathematics and Statistics
Original Paper
Small mammals
Trees
cycle rank
decycling number
upper-embeddable
Graph
decycling set
05C15
ISSN:0911-0119, 1435-5914
Online-Zugang:Volltext
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Zusammenfassung:The decycling number of a graph G , denoted by ∇ ( G ) , is the smallest number of vertices whose removal results in an acyclic subgraph of G . A decycling set S of G with ∇ ( G ) vertices is said to be a ∇ -set. For any connected loopless 4-regular graph G on n vertices, it is shown that n + 1 3 ≤ ∇ ( G ) ≤ n + 1 2 , and presents a necessary and sufficient condition for the two equalities, respectively. Moreover, for any ∇ -set S of G , it is also shown that if G - S is a tree and ∇ ( G ) = n + 1 2 or n 2 , then G is upper-embeddable. Meanwhile, there exists a Xuong-tree T X of G such that vertices of S are leaves of T X .
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0911-0119
1435-5914
DOI:10.1007/s00373-025-02995-6