Bipartite Graphs of Small Readability
- 1.1k Downloads
Abstract
We study a parameter of bipartite graphs called readability, introduced by Chikhi et al. (Discrete Applied Mathematics 2016) and motivated by applications of overlap graphs in bioinformatics. The behavior of the parameter is poorly understood. The complexity of computing it is open and it is not known whether the decision version of the problem is in NP. The only known upper bound on the readability of a bipartite graph (Braga and Meidanis, LATIN 2002) is exponential in the maximum degree of the graph. Graphs that arise in bioinformatic applications have low readability. In this paper we focus on graph families with readability o(n), where n is the number of vertices. We show that the readability of n-vertex bipartite chain graphs is between \(\varOmega (\log n)\) and \(\mathcal {O}(\sqrt{n})\). We give an efficiently testable characterization of bipartite graphs of readability at most 2 and completely determine the readability of grids, showing in particular that their readability never exceeds 3. As a consequence, we obtain a polynomial-time algorithm to determine the readability of induced subgraphs of grids. One of the highlights of our techniques is the appearance of Euler’s totient function in the proof of the upper bound on the readability of bipartite chain graphs. We also develop a new technique for proving lower bounds on readability, which is applicable to dense graphs with a large number of distinct degrees.
Notes
Acknowledgments
The result of Sect. 3.1 was discovered with the help of The On-Line Encyclopedia of Integer Sequences ® [22]. This work has been supported in part by NSF awards DBI-1356529, CCF-1439057, IIS-1453527, and IIS-1421908 to P.M. and by the Slovenian Research Agency (I0-0035, research program P1-0285 and research projects N1-0032, J1-6720, and J1-7051) to M.M. The authors S.R. and N.V. were supported by NSF grant CCF-1422975 to S.R. The author N.V. was also supported by Pennsylvania State University College of Engineering Fellowship, PSU Graduate Fellowship, and by NSF grant IIS-1453527 to P.M. The main idea of the proof of Lemma 4 was developed by V.J. in his undergraduate final project paper [13] at the University of Primorska.
References
- 1.Błażewicz, J., Formanowicz, P., Kasprzak, M., Kobler, D.: On the recognition of de Bruijn graphs and their induced subgraphs. Discrete Math. 245(1), 81–92 (2002)MathSciNetCrossRefGoogle Scholar
- 2.Błażewicz, J., Formanowicz, P., Kasprzak, M., Schuurman, P., Woeginger, G.J.: DNA sequencing, eulerian graphs, and the exact perfect matching problem. In: Goos, G., Hartmanis, J., van Leeuwen, J., Kučera, L. (eds.) WG 2002. LNCS, vol. 2573, pp. 13–24. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-36379-3_2CrossRefGoogle Scholar
- 3.Blazewicz, J., Hertz, A., Kobler, D., de Werra, D.: On some properties of DNA graphs. Discrete Appl. Math. 98(1), 1–19 (1999)MathSciNetCrossRefGoogle Scholar
- 4.Braga, M.D.V., Meidanis, J.: An algorithm that builds a set of strings given its overlap graph. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 52–63. Springer, Heidelberg (2002)CrossRefGoogle Scholar
- 5.Chikhi, R., Jovicic, V., Kratsch, S., Medvedev, P., Milanic, M., Raskhodnikova, S., Varma, N.: Bipartite graphs of small readability. CoRR (2018). http://arxiv.org/abs/1805.04765
- 6.Chikhi, R., Medvedev, P., Milanič, M., Raskhodnikova, S.: On the readability of overlap digraphs. Discrete Appl. Math. 205, 35–44 (2016)MathSciNetCrossRefGoogle Scholar
- 7.Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Math. 86(1–3), 165–177 (1990)MathSciNetCrossRefGoogle Scholar
- 8.Díaz, J., Penrose, M.D., Petit, J., Serna, M.J.: Approximating layout problems on random geometric graphs. J. Algorithms 39(1), 78–116 (2001)MathSciNetCrossRefGoogle Scholar
- 9.Gevezes, T.P., Pitsoulis, L.S.: Recognition of overlap graphs. J. Comb. Optim. 28(1), 25–37 (2014)MathSciNetCrossRefGoogle Scholar
- 10.Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. The Clarendon Press, Oxford University Press, New York (1979)zbMATHGoogle Scholar
- 11.Idury, R.M., Waterman, M.S.: A new algorithm for DNA sequence assembly. J. Comput. Biol. 2(2), 291–306 (1995)CrossRefGoogle Scholar
- 12.Itai, A., Papadimitriou, C.H., Szwarcfiter, J.L.: Hamilton paths in grid graphs. SIAM J. Comput. 11(4), 676–686 (1982)MathSciNetCrossRefGoogle Scholar
- 13.Jovičić, V.: Readability of digraphs and bipartite graphs (2016), final project paper. University of Primorska, Faculty of Mathematics, Natural Sciences and Information Technologies, Koper, Slovenia (2016). https://arxiv.org/abs/1612.07113
- 14.Kasprzak, M.: Classification of de Bruijn-based labeled digraphs. Discrete Appl. Math. 234, 86–92 (2016)MathSciNetCrossRefGoogle Scholar
- 15.Li, X., Zhang, H.: Characterizations for some types of DNA graphs. J. Math. Chem. 42(1), 65–79 (2007)MathSciNetCrossRefGoogle Scholar
- 16.Li, X., Zhang, H.: Embedding on alphabet overlap digraphs. J. Math. Chem. 47(1), 62–71 (2010)MathSciNetCrossRefGoogle Scholar
- 17.Miller, J.R., Koren, S., Sutton, G.: Assembly algorithms for next-generation sequencing data. Genomics 95(6), 315–327 (2010)CrossRefGoogle Scholar
- 18.Myers, E.W.: The fragment assembly string graph. In: ECCB/JBI, p. 85 (2005)Google Scholar
- 19.Nagarajan, N., Pop, M.: Sequence assembly demystified. Nat. Rev. Genet. 14(3), 157–167 (2013)CrossRefGoogle Scholar
- 20.Pendavingh, R., Schuurman, P., Woeginger, G.J.: Recognizing DNA graphs is difficult. Discrete Appl. Math. 127(1), 85–94 (2003)MathSciNetCrossRefGoogle Scholar
- 21.Simpson, J.T., Durbin, R.: Efficient de novo assembly of large genomes using compressed data structures. Genome Res. 22, 549–556 (2011)CrossRefGoogle Scholar
- 22.Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences (2016). https://oeis.org
- 23.Tarhio, J., Ukkonen, E.: A greedy approximation algorithm for constructing shortest common superstrings. Theoret. Comput. Sci. 57(1), 131–145 (1988)MathSciNetCrossRefGoogle Scholar
- 24.West, D.B.: Introduction to Graph Theory. Prentice Hall Inc., Upper Saddle River (1996)zbMATHGoogle Scholar