### 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 (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.

Original language | English (US) |
---|---|

Pages (from-to) | 402-415 |

Number of pages | 14 |

Journal | Theoretical Computer Science |

Volume | 806 |

DOIs | |

State | Accepted/In press - Jan 1 2019 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Theoretical Computer Science*,

*806*, 402-415. https://doi.org/10.1016/j.tcs.2019.07.022

}

*Theoretical Computer Science*, vol. 806, pp. 402-415. https://doi.org/10.1016/j.tcs.2019.07.022

**Bipartite graphs of small readability.** / Chikhi, Rayan; Jovičić, Vladan; Kratsch, Stefan; Medvedev, Paul; Milanič, Martin; Raskhodnikova, Sofya; Varma, Nithin.

Research output: Contribution to journal › Article

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

PY - 2019/1/1

Y1 - 2019/1/1

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.

UR - http://www.scopus.com/inward/record.url?scp=85069802213&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85069802213&partnerID=8YFLogxK

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 -