Bipartite Graphs of Small Readability
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.
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