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A new formula for the decycling number of regular graphs. (English) Zbl 1370.05105
Summary: The decycling number \(\nabla(G)\) of a graph \(G\) is the smallest number of vertices which can be removed from \(G\) so that the resultant graph contains no cycle. A decycling set containing exactly \(\nabla(G)\) vertices of \(G\) is called a \(\nabla\)-set. For any decycling set \(S\) of a \(k\)-regular graph \(G\), we show that \(| S | = \frac{\beta(G) + m(S)}{k - 1}\), where \(\beta(G)\) is the cycle rank of \(G\), \(m(S) = c + | E(S) | - 1\) is the margin number of \(S\), \(c\) and \(| E(S) |\) are, respectively, the number of components of \(G - S\) and the number of edges in \(G [S]\). In particular, for any \(\nabla\)-set \(S\) of a 3-regular graph \(G\), we prove that \(m(S) = \xi(G)\), where \(\xi(G)\) is the Betti deficiency of \(G\). This implies that the decycling number of a 3-regular graph \(G\) is \(\frac{\beta(G) + \xi(G)}{2}\). Hence \(\nabla(G) = \lceil \frac{\beta(G)}{2} \rceil\) for a 3-regular upper-embeddable graph \(G\), which concludes the results in [L. Gao et al., Discrete Appl. Math. 181, 297–300 (2015; Zbl 1304.05082); E. Wei and Y. Li, Acta Math. Sin., Chin. Ser. 56, No. 2, 211–216 (2013; Zbl 1289.05107)] and solves two open problems posed by S. Bau and L. W. Beineke [Australas. J. Comb. 25, 285–298 (2002; Zbl 0994.05079)]. Considering an algorithm by M. Furst, J. L. Gross and L. A. McGeoch [“Finding a maximum genus graph imbedding”, J. Assoc. Comput. Mach. 35, No. 3, 523–534 (1988; doi:10.1145/44483.44485)], there exists a polynomial time algorithm to compute \(Z(G)\), the cardinality of a maximum nonseparating independent set in a 3-regular graph \(G\), which solves an open problem raised by E. Speckenmeyer [J. Graph Theory 12, No. 3, 405–412 (1988; Zbl 0657.05042)]. As for a 4-regular graph \(G\), we show that for any \(\nabla\)-set \(S\) of \(G\), there exists a spanning tree \(T\) of \(G\) such that the elements of \(S\) are simply the leaves of \(T\) with at most two exceptions providing \(\nabla(G) = \lceil \frac{\beta(G)}{3} \rceil\). On the other hand, if \(G\) is a loopless graph on \(n\) vertices with maximum degree at most 4, then \[ \nabla(G) \leq \begin{cases} \frac{n + 1}{2}, & \text{if G is 4-regular}, \\ \frac{n}{2}, & \text{otherwise}. \end{cases}. \] The above two upper bounds are tight, and this makes an extension of a result due to N. Punnim [Thai J. Math. 4, No. 1, 145–161 (2006; Zbl 1156.05317)].

MSC:
05C35 Extremal problems in graph theory
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