Abstract
The determining number of a graph \(G = (V,E)\) is the minimum cardinality of a set \(S\subseteq V\) such that pointwise stabilizer of S under the action of Aut(G) is trivial. In this paper, we determine the determining number of generalized Petersen graphs and double generalized Petersen graphs.
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Acknowledgement
The author is thankful to the anonymous referees for their fruitful suggestions. The author also acknowledge the financial support received under the FRPDF grant of Presidency University, Kolkata and DST-SERB-SRG/2019/000475.
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Appendix
Appendix
In this section, we provide two Sage code for confirming the determining sets of generalized Petersen graphs and double generalized Petersen graphs. The codes are given for G(10, 2) and DP(10, 2). Readers may check for determining sets of other members of these two families by suitably editing the values of the parameters (Fig. 1).
Sage Code for finding a determining set for G(10, 2) (left) and DP(10, 2) (right)
It is checked that \(\{u_0,v_1\}\) is a determining set for G(10, 2) and \(\{x_0,v_1\}\) is a determining set for DP(10, 2). The output of both the codes are 1, showing that there exists exactly one automorphism (namely, the identity automorphism) which stabilizes both \(u_0\) and \(v_1\), and \(x_0\) and \(v_1\), respectively.
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Das, A. (2020). Determining Number of Generalized and Double Generalized Petersen Graph. In: Changat, M., Das, S. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2020. Lecture Notes in Computer Science(), vol 12016. Springer, Cham. https://doi.org/10.1007/978-3-030-39219-2_11
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DOI: https://doi.org/10.1007/978-3-030-39219-2_11
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