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 $\mathrm{\Omega }(\log n)$ 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 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.

我们研究了由Chikhi等人引入的称为“可读性”的二部图参数。（Discrete Applied Mathematics，2016）和重叠图在生物信息学中的应用所激发。参数的行为了解得很少。计算的复杂性是开放的，并且不知道问题的决策版本是否在NP中。二部图的可读性的唯一已知上限（根据Braga和Meidanis的著作，LATIN 2002）在图的最大程度上是指数级的。

生物信息学应用程序中出现的图形的可读性较低。在本文中，我们关注具有可读性的图族$Ø（ñ）$，其中n是顶点数。我们显示n-顶点二分链图的可读性介于$\mathrm{\Omega }（\mathrm{日志}ñ）$ 和 $\mathcal{Ø}（\sqrt{ñ}）$。我们给出了一个可读性最高为2的二部图的有效测试表征，并完全确定了网格的可读性，特别是表明它们的可读性从未超过3。因此，我们获得了多项式时间算法来确定诱导子图的可读性。网格。我们的技术的亮点之一是在分析二分链图的可读性时出现了Euler的totient函数。我们还开发了一种用于证明可读性下限的新技术，该技术适用于具有大量不同程度的密集图。