A characterization of horizontal visibility graphs and combinatorics on words

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.

💡 Research Summary

The paper investigates Horizontal Visibility Graphs (HVGs), a class of graphs constructed from ordered sequences of non‑negative real numbers. For a given sequence (X = (x_1, x_2, \dots, x_n)), two indices (i < j) are connected by an edge if every intermediate value (x_k) (with (i < k < j)) is strictly smaller than (\min(x_i, x_j)). This simple geometric rule captures the “visibility” of points when a horizontal line is drawn between them.

The central theoretical contribution is a complete structural characterization: a graph is an HVG iff it is outerplanar and it contains a Hamiltonian path. Outerplanarity means the graph can be embedded in the plane without edge crossings such that all vertices lie on the outer face. The Hamiltonian path condition reflects the preservation of the original ordering of the data points; the path follows the sequence order and visits every vertex exactly once. The authors prove both directions. First, they show that any HVG can be drawn on a circle with vertices placed in their natural order; the visibility rule guarantees that no two edges cross, establishing outerplanarity, and the natural order itself yields a Hamiltonian path. Conversely, given an outerplanar graph with a Hamiltonian path, they construct a suitable real‑valued sequence by assigning heights that respect the planar embedding, thereby reproducing the graph as an HVG.

From this characterization, a linear‑time recognition algorithm follows immediately. By scanning the input sequence once and maintaining a stack of “currently visible” vertices, one can add or delete edges exactly as the visibility condition dictates. Each vertex is pushed and popped at most once, leading to an overall (O(n)) time complexity, a substantial improvement over previously known quadratic or logarithmic‑factor algorithms.

The paper then adopts a combinatorial viewpoint by interpreting the ordered set as a word over an alphabet. Each real value is mapped to a letter, producing a word (w = w_1 w_2 \dots w_n). A pair ((i,j)) with (i<j) is defined as a visible pair if every intermediate letter (w_k) (for (i<k<j)) is lexicographically smaller than both (w_i) and (w_j). This definition is exactly equivalent to the edge condition in the HVG, establishing a bijection between edges of the graph and visible pairs in the word.

Using this bijection, the authors explore several subfamilies of HVGs.

• Monotone words (strictly increasing or decreasing) generate complete graphs (K_n) because every pair is visible.
• Words with constrained peaks and valleys give rise to trees or forests, depending on how many local maxima/minima are allowed.
• Pattern‑avoiding words (e.g., avoiding the pattern 132) correspond to non‑crossing matchings and Catalan‑type structures.

For each subfamily, the paper relates classical combinatorial statistics—such as the number of ascents, descents, peaks, or runs—to graph properties like degree distribution, number of edges, and connectivity. By employing generating functions and analytic combinatorics, the authors derive the expected number of edges in a random HVG. Specifically, they show that for a uniformly random word of length (n), the average edge count satisfies

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