A characterization of horizontal visibility graphs and combinatorics on words

Gregory Gutin, T. Mansour, S. Severini

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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, \emph{et al.}, \emph{Phys. Rev. E}
\textbf{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, \emph{Discrete
Math.}, \textbf{204} (1999) 203-229]. Our characterization of HVGs implies a
linear time recognition algorithm.\textbf{ }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.
Original languageEnglish
Pages (from-to)2421-2428
JournalPhysica A: Statistical Mechanics and its Applications
Volume390
Publication statusPublished - 2011

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