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In this paper we introduce the horizon visibility graph, a simple extensionto the popular horizontal visibility graph representation of a time series, andshow that it possesses a rigorous mathematical foundation in computationalalgebraic topology. This fills a longstanding gap in the literature on thehorizontal visibility approach to nonlinear time series analysis which, despitea suite of successful applications across multiple domains, lacks a formalsetting in which to prove general properties and develop natural extensions.The main finding is that horizon visibility graphs are dual to merge treesarising naturally over a filtered complex associated to a time series, whilehorizontal visibility graphs are weak duals of these trees. Immediateconsequences include availability of tree-based reconstruction theorems,connections to results on the statistics of self-similar trees, and relationsbetween visibility graphs and the emerging field of applied persistenthomology.

Horizon Visibility Graphs and Time Series Merge Trees are Dual