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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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##### References:
 [1] M. Albertson, D. Berman, The acyclic chromatic number, Congr. Numer., 17(1976), 51-69.. · Zbl 0344.05116 [2] S. Bau, L. Beineke, The decycling number of graphs, Australas. J. Combin., 25(2002), 285-298.. · Zbl 0994.05079 [3] S. Bau, N. Wormald, S. Zhou, Decycling numbers of random regular graphs, Random Structures Algorithms, 21(2002), no. 3-4, 397-413.. · Zbl 1012.05099 [4] L. Beineke, R. Vandell, Decycling graphs, J. Graph Theory, 25(1997), no. 1, 59-77.. · Zbl 0870.05033 [5] R. Diestel, Graph Theory, Springer-Verlag, New York, 1997.. [6] P. Erdös, M. Saks, V. Sós, Maximum induced trees in graphs, J. Combin. Theory Ser. B, 41(1986), 61-79.. [7] R.Focardi, F.Luccio, D.Peleg, Feedback vertex set in hypercubes, Inform.Process.Lett., 76(2000), no.1-2, 1-5.. · Zbl 1338.68218 [8] M.Furst, J.Gross, L.Mcgeoch, Finding a maximum genus graph imbedding, J.Assoc.Comput.Mach, 35(1988), no.3, 523-534.. [9] L.Gao, X.Xu, J.Wang, D.Zhu, Y.Yang, The decycling number of generalized Petersen graphs, Discrete Appl. Math., 181(2015), 297C300.. · Zbl 1304.05082 [10] F. Harary, Graph Theory, Academic Press, New York, 1967.. [11] G.Kirchhoff, Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme geführt wird, Ann. Phys. Chem., 72(1847), 497-508.. [12] R.Karp, Reducibility among combinatorial problems, Complexity of computer computations, (Proc. Sympos., IBM Thomas J.Watson Res.Center, Yorktown Heights, N.Y., 1972), 85C103.Plenum, New York, 1972.. · Zbl 1187.90014 [13] Y.Liu, The maximum orientable genus of a graph, Sci.Sinica, Special Issue on Math., 1979, Special Issue II on Math., 41-55.. [14] D.Ma, Embedding of graphs on surface and crossing numbers, Master thesis, Shanghai, East China Normal University, 2004.. [15] D.Pike, Y.Zou, Decycling cartesian products of two cycles, SIAM J.Discrete Math., 19(2005), no.3, 651-663.. · Zbl 1096.05030 [16] N. Punnim, The decycling number of cubic graphs, Combinatorial Geometry and Graph Theory, 141-145, Lecture Notes in Comput. Sci., 3330, Springer, Berlin, 2005.. · Zbl 1117.05022 [17] N.Punnim, The decycling number of cubic planar graphs, Discrete Geometry, Combinatorics and Graph Theory, 149-161, Lecture Notes in Comput. Sci., 4381, Springer, Berlin, 2007.. · Zbl 1149.05315 [18] N. Punnim, The decycling number of regular graphs, Thai J. Math., 4(2006), 145-161.. · Zbl 1156.05317 [19] H. Ren, S. Long, The decycling number and maximum genus of cubic graphs. (Submitted). · Zbl 1393.05166 [20] E.Speckenmeyer, On feedback vertex sets and nonseparating independent sets in cubic graphs, J. Graph Theory, 12(1988), no. 3, 405-412.. · Zbl 0657.05042 [21] E. Wei, Y. Liu, Z. Li, Decycling number of circular graphs, ISORA’09, (2009), 387-393.. [22] E.Wei, Y.Li, Decycling number and upper-embeddiblity of generalized Petersen graphs, Acta Math. Sinica(Chin. Ser.), 56(2013), no. 2, 211-216.. · Zbl 1289.05107 [23] N.Xuong, How to determine the maximum genus of a graph, J.Combin.Theory Ser.B, 26(1979), 217-225.. · Zbl 0403.05035
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