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The decycling number of generalized Petersen graphs. (English) Zbl 1304.05082
Summary: A subset \(F \subset V(G)\) is called a decycling set if the subgraph \(G - F\) is acyclic. The minimum cardinality of a decycling set is called the decycling number of \(G\), which is proposed first by L. W. Beineke and R. C. Vandell [J. Graph Theory 25, No. 1, 59–77 (1997; Zbl 0870.05033)]. We use \(\nabla(P_{n, k})\) to denote the decycling number of the generalized Petersen graphs \(P_{n, k}\). This paper proves that
\[ \nabla(P_{n, k}) = \begin{cases} \lceil \frac{n + 1}{2} \rceil,&\text{if } n \neq 2 k, \\ \lceil \frac{k + 1}{2} \rceil, &\text{if } n = 2 k.\end{cases} \]

05C38 Paths and cycles
05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.)
Full Text: DOI
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