Abstract
Orbits of graphs under local complementation (LC) and edge local complementation (ELC) have been studied in several different contexts. For instance, there are connections between orbits of graphs and error-correcting codes. We define a new graph class, ELC-preserved graphs, comprising all graphs that have an ELC orbit of size one. Through an exhaustive search, we find all ELC-preserved graphs of order up to 12 and all ELC-preserved bipartite graphs of order up to 16. We provide general recursive constructions for infinite families of ELC-preserved graphs, and show that all known ELC-preserved graphs arise from these constructions or can be obtained from Hamming codes. We also prove that certain pairs of ELC-preserved graphs are LC equivalent. We define ELC-preserved codes as binary linear codes corresponding to bipartite ELC-preserved graphs, and study the parameters of such codes.
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References
Kotzig A.: Eulerian lines in finite 4-valent graphs and their transformations. In: Theory of Graphs. Proceedings of the Colloquium, Tihany, 1966, pp. 219–230. Academic Press, New York (1968).
de Fraysseix H.: Local complementation and interlacement graphs. Discret. Math. 33(1), 29–35 (1981).
Fon-der Flaas D.G.: On local complementations of graphs. In: Combinatorics, Eger, 1987. Colloquium of the Mathematical Society of János Bolyai, vol. 52, pp. 257–266. North-Holland, Amsterdam (1988).
Bouchet A.: Graphic presentations of isotropic systems. J. Comb. Theory Ser. B 45(1), 58–76 (1988).
Hein M., Eisert J., Briegel H.J.: Multi-party entanglement in graph states. Phys. Rev. A 69(6), 062311 (2004).
Van den Nest M., Dehaene J., De Moor B.: Graphical description of the action of local Clifford transformations on graph states. Phys. Rev. A 69(2), 022316 (2004).
Calderbank A.R., Rains E.M., Shor P.M., Sloane N.J.A.: Quantum error correction via codes over GF(4). IEEE Trans. Inf. Theory 44(4), 1369–1387 (1998).
Danielsen L.E., Parker M.G.: On the classification of all self-dual additive codes over GF(4) of length up to 12. J. Comb. Theory Ser. A 113(7), 1351–1367 (2006).
Riera C., Parker M.G.: On pivot orbits of Boolean functions. In: Proceedings of the Fourth International Workshop on Optimal Codes and Related Topics, pp. 248–253. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia (2005).
Riera C., Parker M.G.: Generalised bent criteria for Boolean functions (I). IEEE Trans. Inf. Theory 52(9), 4142–4159 (2006).
Aigner M., van der Holst H.: Interlace polynomials. Linear Algebra Appl. 377, 11–30 (2004).
Arratia R., Bollobás B., Sorkin G.B.: The interlace polynomial of a graph. J. Comb. Theory Ser. B 92(2), 199–233 (2004).
Arratia R., Bollobás B., Coppersmith D., Sorkin G.B.: Euler circuits and DNA sequencing by hybridization. Discret. Appl. Math. 104, 63–96 (2000).
Danielsen L.E., Parker M.G.: Interlace polynomials: enumeration, unimodality, and connections to codes. Discret. Appl. Math. 158(6), 636–648 (2010).
Bouchet A.: Circle graph obstructions. J. Comb. Theory Ser. B 60(1), 107–144 (1994).
Geelen J., Oum S.: Circle graph obstructions under pivoting. J. Graph Theory 61(1), 1–11 (2009).
Danielsen L.E., Parker M.G.: Edge local complementation and equivalence of binary linear codes. Des. Codes Cryptogr. 49, 161–170 (2008).
Knudsen J.G., Riera C., Parker M.G., Rosnes E.: Adaptive soft-decision iterative decoding using edge local complementation. In: Second International Castle Meeting on Coding Theory and Applications—ICMCTA 2008. Lecture Notes in Computer Science, vol. 5228, pp. 82–94. Springer, Berlin (2008).
Knudsen J.G., Riera C., Danielsen L.E., Parker M.G., Rosnes E.: Iterative decoding on multiple Tanner graphs using random edge local complementation. In: Proceedings of the IEEE International Symposium on Information Theory, Seoul, pp. 899–903 (2009).
Knudsen J.G., Riera C., Danielsen L.E., Parker M.G., Rosnes E.: Random edge-local complementation with applications to iterative decoding of HDPC codes. Reports in Informatics 395, University of Bergen, August (2010).
Knudsen J.G., Riera C., Danielsen L.E., Parker M.G., Rosnes E.: Improved adaptive belief propagation decoding using edge-local complementation. In: Proceedings of the IEEE International Symposium on Information Theory, Austin, pp. 774–778 (2010).
Halford T.R., Chugg K.M.: Random redundant iterative soft-in soft-out decoding. IEEE Trans. Commun. 56(4), 513–517 (2008).
Curtis R.T.: On graphs and codes. Geom. Dedicata 41(2), 127–134 (1992).
Parker M.G., Rijmen V.: The quantum entanglement of binary and bipolar sequences. In: Sequences and Their Applications—SETA ’01. Discrete Mathematics and Theoretical Computer Science, pp. 296–309. Springer, London (2002).
Bollobás B.: Modern Graph Theory. Graduate Texts in Mathematics, vol. 184. Springer, New York (1998).
Ellis-Monaghan J.A., Sarmiento I.: Distance hereditary graphs and the interlace polynomial. Comb. Probab. Comput. 16(6), 947–973 (2007).
Pless V.S., Huffman W.C. (eds.): Handbook of Coding Theory. North-Holland, Amsterdam (1998).
Bilous R.T., van Rees G.H.J.: An enumeration of binary self-dual codes of length 32. Des. Codes Cryptogr. 26, 61–86 (2002).
Bilous R.T.: Enumeration of the binary self-dual codes of length 34. J. Comb. Math. Comb. Comput. 59, 173–211 (2006).
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This research was supported by the Research Council of Norway. The authors would like to thank the anonymous reviewers for providing useful suggestions and corrections that improved the quality of the manuscript.
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Communicated by W. H. Haemers.
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Danielsen, L.E., Parker, M.G., Riera, C. et al. On graphs and codes preserved by edge local complementation. Des. Codes Cryptogr. 74, 601–621 (2015). https://doi.org/10.1007/s10623-013-9876-6
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DOI: https://doi.org/10.1007/s10623-013-9876-6