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Horizontal visibility graphs (HVGs) are graphs constructed in correspondencewith number sequences that have been introduced and explored recently in thecontext of graph-theoretical time series analysis. In most of the cases simplemeasures based on the degree sequence (or functionals of these such asentropies over degree and joint degree distributions) appear to be highlyinformative features for automatic classification and provide nontrivialinformation on the associated dynam- ical process, working even better thanmore sophisticated topological metrics. It is thus an open question why theseseemingly simple measures capture so much information. Here we prove that,under suitable conditions, there exist a bijection between the adjacency matrixof an HVG and its degree sequence, and we give an explicit construction of suchbijection. As a consequence, under these conditions HVGs are unigraphs and thedegree sequence fully encapsulates all the information of these graphs, therebygiving a plausible reason for its apparently unreasonable effectiveness.

Canonical Horizontal Visibility Graphs are uniquely determined by their degree sequence