  Horizontal visibility graphs (HVGs) are graphs constructed in correspondence
with number sequences that have been introduced and explored recently in the
context of graph-theoretical time series analysis. In most of the cases simple
measures based on the degree sequence (or functionals of these such as
entropies over degree and joint degree distributions) appear to be highly
informative features for automatic classification and provide nontrivial
information on the associated dynam- ical process, working even better than
more sophisticated topological metrics. It is thus an open question why these
seemingly simple measures capture so much information. Here we prove that,
under suitable conditions, there exist a bijection between the adjacency matrix
of an HVG and its degree sequence, and we give an explicit construction of such
bijection. As a consequence, under these conditions HVGs are unigraphs and the
degree sequence fully encapsulates all the information of these graphs, thereby
giving a plausible reason for its apparently unreasonable effectiveness.
