  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.
