# Linear Algorithm for a Cyclic Graph Transformation

- 4 Downloads

## Abstract

We propose a linear time and linear space algorithm that constructs a minimal (in the total cost) sequence of operations required to transform a directed graph consisting of disjoint cycles into any graph of the same type. The following operations are allowed: double-cut-and-join of vertices and insertion or deletion of a connected fragment of edges; the latter two operations have the same cost. We present a complete proof that the algorithm finds the corresponding minimum. The problem is a nontrivial particular case of the general problem of transforming a graph into another, which in turn is an instance of a hard optimization problem in graphs. In this *general* problem, which is known to be NP-complete, each vertex of a graph is of degree 1 or 2, edges with the same name may repeat unlimitedly, and operations belong to a well-known list including the above-mentioned operations. We describe our results for the general problem, but the proof is given for the cyclic case only.

## Keywords and phrases

graph cycle graph rearrangement operation cost combinatorial problem optimization in graphs linear algorithm## Preview

Unable to display preview. Download preview PDF.

## References

- 1.K. Yu. Gorbunov and V. A. Lyubetsky, “A linear algorithmfor the shortest transformation of graphs with different operation costs,” J. Commun. Technol. Electron.
**62**, 653–662 (2017). doi 10.1134/S1064226917060092CrossRefGoogle Scholar - 2.K. Yu. Gorbunov and V. A. Lyubetsky, “Linear algorithm for minimal rearrangement of structures,” Probl. Inform. Transmiss.
**53**, 55–72 (2017). doi 10.1134/S0032946017010057MathSciNetCrossRefzbMATHGoogle Scholar - 3.K. Yu. Gorbunov and V. A. Lyubetsky, “A linear algorithm of graph reconfiguration,” Autom. Remote Control, No. 12 (2018, in press).Google Scholar
- 4.K. Yu. Gorbunov, R. A. Gershgorin, and V. A. Lyubetsky, “Rearrangement and inference of chromosome structures,” Mol. Biol. (Moscow)
**49**, 327–338 (2015). doi 10.1134/S0026893315030073CrossRefGoogle Scholar - 5.V. A. Lyubetsky, R. A. Gershgorin, A. V. Seliverstov, and K. Yu. Gorbunov, “Algorithms for reconstruction of chromosomal structures,” BMC Bioinform. 17, 40. 1–40.
**23**(2016). doi 10.1186/s12859-016-0878-zGoogle Scholar - 6.V. A. Lyubetsky, R. A. Gershgorin, and K. Yu. Gorbunov, “Chromosome structures: reduction of certain problems with unequal gene content and gene paralogs to integer linear programming,” BMC Bioinform. 18, 537. 1–537.
**18**(2017). doi 10.1186/s12859-017-1944-xGoogle Scholar - 7.Z. Yin, J. Tang, S. W. Schaeffer, and D. A. Bader, “Exemplar or matching: modeling DCJ problems with unequal content genome data,” J. Combinat. Optimiz.
**32**, 1165–1181 (2016). doi 10.1007/s10878-015-9940-4MathSciNetCrossRefzbMATHGoogle Scholar - 8.
*Models and Algorithms for Genome Evolution, Ed. by C. Chauve, N. El-Mabrouk, and E. Tannier, Comput. Biol. Series*(Springer, London, 2013).Google Scholar - 9.K. Yu. Gorbunov and V. A. Lyubetsky, “The minimum-cost transformation of graphs,” Dokl. Math.
**96**, 503–505 (2017). doi 10.1134/S1064562417050313MathSciNetCrossRefzbMATHGoogle Scholar - 10.R. A. Gershgorin, K. Yu. Gorbunov, O. A. Zverkov, L. I. Rubanov, A. V. Seliverstov, and V. A. Lyubetsky, “Highly conserved elements and chromosome structure evolution in mitochondrial genomes in ciliates,” Life 7, 9. 1–9. 11 (2017). doi 10.3390/life7010009Google Scholar
- 11.M. D. V. Braga, E. Willing and J. Stoye, “Double cut and join with insertions and deletions,” J. Comput. Biol.
**18**, 1167–1184 (2011). doi 10.1089/cmb. 2011. 0118MathSciNetCrossRefGoogle Scholar - 12.P. H. da Silva, R. Machado, S. Dantas, and M. D. V. Braga, “DCJ-indel and DCJ-substitution distances with distinct operation costs,” Algorithms Mol. Biol. 8, 21. 1–21. 15 (2013). doi 10.1186/1748-7188-8-21Google Scholar
- 13.P. E. C. Compeau, “DCJ-indel sorting revisited,” Algorithms Mol. Biol. 8, 6. 1–6. 9 (2013). doi 10.1186/1748-7188-8-6Google Scholar
- 14.P. E. C. Compeau, “A generalized cost model for DCJ-indel sorting,” in
*Proceedings of 14th International Workshop on Algorithms in Bioinformatics, Wroclaw, Poland, Sept. 8–10, 2014, Lect. Notes Comput. Sci.***8701**, 38–51 (2014). doi 10.1007/978-3-662-44753-6_4MathSciNetGoogle Scholar - 15.S. Hannenhalli and P. A. Pevzner, “Transforming men into mice (polynomial algorithm for genomic distance problem),” in
*Proceedings of the 36th Annual Symposiumon Foundations of Computer Science—FOCS 1995, Milwaukee, USA, Oct. 23–25*, 1995, pp. 581–592.Google Scholar - 16.G. Li, X. Qi, X. Wang, and B. Zhu, “A linear-time algorithm for computing translocation distance between signed genomes,” in
*Proceedings of 15th Annual Symposium on Combinatorial Pattern Matching—CPM 2004, July 5–7, 2004, Istanbul, Turkey, Lect. Notes Comput. Sci.***3109**, 323–332 (2004). doi 10.1007/978-3-540-27801-6_24zbMATHGoogle Scholar - 17.A. Bergeron, J. Mixtacki, and J. Stoye, “A new linear time algorithm to compute the genomic distance via the double cut and join distance,” Theor. Comput. Sci.
**410**, 5300–5316 (2009). doi 10.1016/j. tcs. 2009. 09. 008MathSciNetCrossRefzbMATHGoogle Scholar - 18.A. Bergeron, J. Mixtacki, and J. Stoye, “A unifying view of genome rearrangements,” in
*Proc. of 6th InternationalWorkshop on Algorithms in Bioinformatics, Zurich, Switzerland, Sept. 8–10, 2006, Lect. Notes Bioinform.***4175**, 163–173 (2006). doi 10.1007/11851561_16MathSciNetGoogle Scholar - 19.M. A. Alekseyev and P. A. Pevzner, “Multi-break rearrangements and chromosomal evolution,” Theor. Comput. Sci.
**395**, 193–202 (2008). doi 10.1016/j. tcs. 2008. 01. 013MathSciNetCrossRefzbMATHGoogle Scholar