A characterization of horizontal visibility graphs and combinatorics on words ## AbstractAn 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.
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