JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2015, Vol. 50 ›› Issue (12): 130-136.doi: 10.6040/j.issn.1671-9352.0.2014.567

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Solving the algebraic model of k-dominating sets of graph by Grbner basis

YIN Jie-jie   

  1. Department of Applied Mathematics of Information and Technology College, Hainan University, Haikou 570228, Hainan, China
  • Received:2014-12-22 Revised:2015-11-18 Online:2015-12-20 Published:2015-12-23

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Abstract: Let G be a simple connected graph of n vertices. It is shown that finding all dominating sets of G is equivalent to finding all 0-1 solutions of a system of multivariate polynomial equations; furthermore, it is shown that this polynomial equation model can be modified to give a polynomial equation model for finding dominating sets of k vertices (i.e. k-dominating sets) of G, and that such a model can be solved by using the Grbner basis method. Consequently, a feasible way of finding minimal dominating sets and the dominating number of G is obtained. The numerical example is presented to illustrate the effectiveness of this algebraic computational method.

Key words: k-dominating set, Grbner basis, minimal dominating set, dominating number, Graph

CLC Number: 

  • O157. 5
[1] ADAMS W, LOUSTAUNAUS P. An introduction to Grbner bases[M]. Washington, D C: American Mathematical Society, 1994:118-276.
[2] Sk Md Abu Nayeem, PAL Madhumangal. Genetic algorithmic approach to find the maximum weight independent set of a graph[J]. Journal of Applied Mathematics and Computing, 2007, 25(1):217-219.
[3] 熊雪玮,赵志琴. 图的k-独立集与Grbner 基求解[J].工程数学学报,2012, 29(5):696-702. XIONG Xuewei, ZHAO Zhiqing. Solving the k-independent sets of graph by Grbner basis[J]. Chinese Journal Engineering Mathematics, 2012, 29(5):696-702.
[4] 熊雪玮.几个图论问题的多项式建模与Grbner 基求解[M]. 海口:海南大学出版社,2012. XIONG Xuewei. The polynomial modeling and Grbner basis solving of several problems in graph theory[M]. Haikou: Hainan University Press, 2012.
[5] MARGULIES S, HICKS I V. An algebraic exploration of dominating sets and Vizing's conjecture[J]. The Electronic Journal of Combinatorics, 2012, 19(2):1-30.
[6] MNUK M. Representing graph properties by polynomial ideals[M]. New York: Computer Algebra in Scientific Computing, 2001:431-444.
[7] BONDY J A, MURTY U S R. Graph theory [M]. Berlin: Springer, 2008: 78-156.
[8] 殷剑宏,吴开亚. 图论及其算法[M]. 合肥:中国科学技术大学出版社,2003:179-185. YIN Jianhong,WU Kaiya. Graph theory and algorithms[M]. Hefei:University of Science and Technology of China Press, 2003:179-185.
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