an:06775857 Zbl 1370.05105 Ren, Han; Yang, Chao; Zhao, Tian-xiao A new formula for the decycling number of regular graphs EN Discrete Math. 340, No. 12, 3020-3031 (2017). 00369921 2017
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05C35 decycling number; margin number; cycle rank; Betti deficiency; regular graphs 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 [\textit{L. Gao} et al., Discrete Appl. Math. 181, 297--300 (2015; Zbl 1304.05082); \textit{E. Wei} and \textit{Y. Li}, Acta Math. Sin., Chin. Ser. 56, No. 2, 211--216 (2013; Zbl 1289.05107)] and solves two open problems posed by \textit{S. Bau} and \textit{L. W. Beineke} [Australas. J. Comb. 25, 285--298 (2002; Zbl 0994.05079)]. Considering an algorithm by \textit{M. Furst}, \textit{J. L. Gross} and \textit{L. A. McGeoch} [Finding a maximum genus graph imbedding'', J. Assoc. Comput. Mach. 35, No. 3, 523--534 (1988; \url{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 \textit{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 \textit{N. Punnim} [Thai J. Math. 4, No. 1, 145--161 (2006; Zbl 1156.05317)]. Zbl 1304.05082; Zbl 1289.05107; Zbl 0994.05079; Zbl 0657.05042; Zbl 1156.05317