On the readability of overlap digraphs

TL;DR

This paper introduces the concept of readability in overlap digraphs, analyzing its behavior as a function of graph size, and provides bounds for various graph families through a graph-theoretic approach.

Contribution

It defines the graph parameter readability, studies its asymptotic behavior, and establishes bounds for different graph classes without relying on string representations.

Findings

Upper and lower bounds on readability for specific graph families

Readability reflects string length in applications like bioinformatics

Asymptotic analysis of readability as graph size increases

Abstract

We introduce the graph parameter readability and study it as a function of the number of vertices in a graph. Given a digraph D, an injective overlap labeling assigns a unique string to each vertex such that there is an arc from x to y if and only if x properly overlaps y. The readability of D is the minimum string length for which an injective overlap labeling exists. In applications that utilize overlap digraphs (e.g., in bioinformatics), readability reflects the length of the strings from which the overlap digraph is constructed. We study the asymptotic behaviour of readability by casting it in purely graph theoretic terms (without any reference to strings). We prove upper and lower bounds on readability for certain graph families and general graphs