TY - JOUR

T1 - Bipartite graphs of small readability

AU - Chikhi, Rayan

AU - Jovičić, Vladan

AU - Kratsch, Stefan

AU - Medvedev, Paul

AU - Milanič, Martin

AU - Raskhodnikova, Sofya

AU - Varma, Nithin

N1 - Funding Information:
The authors are grateful to the two anonymous reviewers for their helpful remarks. The result of Section 3.1 was discovered with the help of The On-Line Encyclopedia of Integer Sequences ® [23] . 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 , N1-0102 , J1-6720 , and J1-7051 ) to M.M. The authors S.R. and N.V. were supported in part by NSF grants CCF-1422975 and CCF-1832228 to S.R. The author N.V. was also supported by Pennsylvania State University College of Engineering Fellowship and Pennsylvania State University Graduate Fellowship and in part by NSF grant IIS-1453527 to P.M. V.J. did most of his work on the paper while he was an undergraduate student at the University of Primorska. The main idea of the proof of Lemma 6 was developed in his final project paper [14] .
Publisher Copyright:
© 2019 Elsevier B.V.

PY - 2020/2/2

Y1 - 2020/2/2

N2 - 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 (following from a work of Braga and Meidanis, LATIN 2002) is exponential in the maximum degree of the graph. Graphs that arise in bioinformatics 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 Ω(logn) and O(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 analysis of 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.

AB - 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 (following from a work of Braga and Meidanis, LATIN 2002) is exponential in the maximum degree of the graph. Graphs that arise in bioinformatics 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 Ω(logn) and O(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 analysis of 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.

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U2 - 10.1016/j.tcs.2019.07.022

DO - 10.1016/j.tcs.2019.07.022

M3 - Article

AN - SCOPUS:85069802213

VL - 806

SP - 402

EP - 415

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -