Bipartite Graphs of Small Readability

TL;DR

This paper investigates the parameter of bipartite graph readability, providing bounds, characterizations, and algorithms for specific graph classes, and introduces new techniques for analyzing readability in dense graphs.

Contribution

It offers the first bounds for bipartite chain graphs, characterizes bipartite graphs with readability at most 2, and determines the readability of grid graphs, advancing understanding of this parameter.

Findings

Readability of bipartite chain graphs is between Ω(log n) and O(√n).

Bipartite graphs with readability at most 2 are characterized and efficiently testable.

Readability of grid graphs is at most 3, with polynomial-time algorithms for induced subgraphs.

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 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 $\Omega(\log n)$ and $O(\sqrt{n})$. 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.