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)
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Bipartite graphs of small readability. / Chikhi, Rayan; Jovičić, Vladan; Kratsch, Stefan; Medvedev, Paul; Milanič, Martin; Raskhodnikova, Sofya; Varma, Nithin.
In: Theoretical Computer Science, Vol. 806, 02.02.2020, p. 402-415.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.
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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 -