Bipartite graphs of small readability

Rayan Chikhi, Vladan Jovičić, Stefan Kratsch, Paul Medvedev, Martin Milanič, Sofya Raskhodnikova, Nithin Varma

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

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 (Formula Presented) and (Formula Presented). 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.

Original languageEnglish (US)
Title of host publicationComputing and Combinatorics - 24th International Conference, COCOON 2018, Proceedings
EditorsDaming Zhu, Lusheng Wang
PublisherSpringer Verlag
Pages467-479
Number of pages13
ISBN (Print)9783319947754
DOIs
StatePublished - Jan 1 2018
Event24th International Conference on Computing and Combinatorics Conference, COCOON 2018 - Qing Dao, China
Duration: Jul 2 2018Jul 4 2018

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume10976 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other24th International Conference on Computing and Combinatorics Conference, COCOON 2018
CountryChina
CityQing Dao
Period7/2/187/4/18

Fingerprint

Bioinformatics
Bipartite Graph
Chain Graph
Graph in graph theory
Polynomials
Euler's phi function
Upper bound
Grid
Discrete mathematics
Induced Subgraph
Applied mathematics
Maximum Degree
Polynomial-time Algorithm
Overlap
Exceed
Lower bound
Distinct
Computing
Vertex of a graph

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Chikhi, R., Jovičić, V., Kratsch, S., Medvedev, P., Milanič, M., Raskhodnikova, S., & Varma, N. (2018). Bipartite graphs of small readability. In D. Zhu, & L. Wang (Eds.), Computing and Combinatorics - 24th International Conference, COCOON 2018, Proceedings (pp. 467-479). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10976 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-94776-1_39
Chikhi, Rayan ; Jovičić, Vladan ; Kratsch, Stefan ; Medvedev, Paul ; Milanič, Martin ; Raskhodnikova, Sofya ; Varma, Nithin. / Bipartite graphs of small readability. Computing and Combinatorics - 24th International Conference, COCOON 2018, Proceedings. editor / Daming Zhu ; Lusheng Wang. Springer Verlag, 2018. pp. 467-479 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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Chikhi, R, Jovičić, V, Kratsch, S, Medvedev, P, Milanič, M, Raskhodnikova, S & Varma, N 2018, Bipartite graphs of small readability. in D Zhu & L Wang (eds), Computing and Combinatorics - 24th International Conference, COCOON 2018, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 10976 LNCS, Springer Verlag, pp. 467-479, 24th International Conference on Computing and Combinatorics Conference, COCOON 2018, Qing Dao, China, 7/2/18. https://doi.org/10.1007/978-3-319-94776-1_39

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

Computing and Combinatorics - 24th International Conference, COCOON 2018, Proceedings. ed. / Daming Zhu; Lusheng Wang. Springer Verlag, 2018. p. 467-479 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10976 LNCS).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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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

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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 (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 (Formula Presented) and (Formula Presented). 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.

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 (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 (Formula Presented) and (Formula Presented). 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.

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Chikhi R, Jovičić V, Kratsch S, Medvedev P, Milanič M, Raskhodnikova S et al. Bipartite graphs of small readability. In Zhu D, Wang L, editors, Computing and Combinatorics - 24th International Conference, COCOON 2018, Proceedings. Springer Verlag. 2018. p. 467-479. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-319-94776-1_39