A characterization of horizontal visibility graphs and combinatorics on words

TL;DR

This paper characterizes horizontal visibility graphs (HVGs) as outerplanar graphs with a Hamilton path, providing a linear time recognition algorithm and exploring their combinatorial properties through words and statistics.

Contribution

It offers a complete characterization of HVGs as outerplanar graphs with a Hamilton path and links them to combinatorial word statistics.

Findings

HVGs are exactly outerplanar graphs with a Hamilton path.

A linear time recognition algorithm for HVGs is developed.

Asymptotic average number of edges in HVGs is determined.

Abstract

An Horizontal Visibility Graph (for short, HVG) is defined in association with an ordered set of non-negative reals. HVGs realize a methodology in the analysis of time series, their degree distribution being a good discriminator between randomness and chaos [B. Luque, et al., Phys. Rev. E 80 (2009), 046103]. We prove that a graph is an HVG if and only if outerplanar and has a Hamilton path. Therefore, an HVG is a noncrossing graph, as defined in algebraic combinatorics [P. Flajolet and M. Noy, Discrete Math., 204 (1999) 203-229]. Our characterization of HVGs implies a linear time recognition algorithm. Treating ordered sets as words, we characterize subfamilies of HVGs highlighting various connections with combinatorial statistics and introducing the notion of a visible pair. With this technique we determine asymptotically the average number of edges of HVGs.