A characterization of horizontal visibility graphs and combinatorics on words

Gregory Gutin, Toufik Mansour, Simone Severini

Research output: Contribution to journalArticlepeer-review


A Horizontal Visibility Graph (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 Luque et al. [B. Luque, L. Lacasa, F. Ballesteros, J. Luque, Horizontal visibility graphs: exact results for random time series, Phys. Rev. E 80 (2009), 046103]. We prove that a graph is an HVG if and only if it is outerplanar and has a Hamilton path. Therefore, an HVG is a noncrossing graph, as defined in algebraic combinatorics Flajolet and Noy [P. Flajolet, M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math., 204 (1999) 203229]. 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.

Original languageEnglish
Pages (from-to)2421-2428
Number of pages8
JournalPhysica A: Statistical Mechanics and its Applications
Issue number12
StatePublished - 15 Jun 2011


  • Networks
  • Time series

ASJC Scopus subject areas

  • Statistics and Probability
  • Condensed Matter Physics


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