Visibility Graphs

• Visibility graphs are geometric-combinatorial constructions that map physical line-of-sight between points, segments, or pixels, enabling analysis in data-rich fields.
• They use specific rules—such as natural, horizontal, and k-visibility—to efficiently connect vertices based on unobstructed views, with algorithms that scale to complex datasets.
• Key insights include rigorous graph-theoretic properties, NP-hard recognition challenges, and applications in time series analysis, image processing, and neural network feature extraction.

Visibility graphs are a geometric-combinatorial construction encoding line-of-sight relationships in point sets, polygonal structures, spatial data arrays, and time series, with deep applications in computational geometry, network science, and signal analysis. At their core, a visibility graph assigns vertices to geometric objects (points, segments, bars, arcs, rectangles, etc.) and places edges according to criteria mimicking physical visibility or unobstructed connection—often requiring that a segment between two vertices contains no intervening object or point. The precise visibility rule and the class of objects considered (points, bars, arcs, polygons, etc.) determines numerous subclasses (point visibility graphs, bar/arc/k-visibility graphs, polygon visibility graphs, terrain visibility graphs, rectangle visibility graphs, transparent rectangle visibility graphs, etc.). This article surveys foundational theories, construction algorithms, graph-theoretic properties, complexity results, practical applications, and open problems.

1. Formal Architectures and Variants

1.1 Natural and Horizontal Visibility Graphs

Let $\{(x_i, y_i)\}_{i=1}^N$ be a sequence of real-valued measurements (time series, spatial scans, or feature map activations) indexed such that $x_i < x_{i+1}$. The natural visibility graph (NVG) connects $(x_i, y_i)$ and $(x_j, y_j)$ ($i<j$) iff for every intermediate $k$ ($i<k<j$),

$$y_k < y_j + (y_i - y_j)\frac{x_j - x_k}{x_j - x_i}\,.$$

This ensures the line of sight between $(x_i, y_i)$ and $(x_j, y_j)$ is never "obscured" by a higher intermediate point.

The horizontal visibility graph (HVG) is a restrictive variant: $(x_i, y_i)$ and $(x_j, y_j)$ ($i<j$) are linked iff for every $k$ ($i<k<j$),

$y_k < \min\{y_i, y_j\}.$

This restricts visibility to horizontal rays above all intermediates. HVGs admit closed combinatorial characterizations and linear-time recognition (Gutin et al., 2010).

1.2 Weighted Visibility Graphs

If $(x_i, y_i)$ and $(x_j, y_j)$ are connected in the NVG, assign to edge $(i, j)$ the weight

$$w_{ij} = \arctan\left(\frac{y_j - y_i}{x_j - x_i}\right) \in (-\tfrac{\pi}{2}, \tfrac{\pi}{2}),$$

interpreted as the viewing angle or slope (Florindo et al., 2021).

1.3 k-Visibility Graphs

Given geometric objects (bars, arcs, circles), the k-visibility graph connects two objects if a sightline joining them intersects at most $k$ other objects. Increasing $k$ allows "seeing through" further occlusions, interpolating between classical (0-visibility) and complete graphs as $k\to\infty$ (Babbitt et al., 2013, Sawhney et al., 2016).

Bar and Arc k-Visibility Definitions

• Bar k-visibility: Vertices correspond to horizontal line segments. Two bars are adjacent iff a vertical sightline joins them through at most $k$ other bars.
• Arc k-visibility: Vertices as concentric arcs; adjacent if a radial sightline intersects up to $k$ additional arcs.

2. Image, Spatial, and Higher-Dimensional Visibility Graphs

Visibility graphs generalize from 1D to spatially extended data.

2D and dD Scalar Fields

Given $I \in \mathbb{R}^{N \times N}$ (image or 2D field), the image visibility graph (IVG) connects pixels $(i,j)$ and $(k,\ell)$ along aligned directions ($i=k$, $j=\ell$, $i-k = j-\ell$, $i-k = \ell-j$), provided all intermediates satisfy the convexity criterion.

• IHVG (FCC extension): Use the HVG rule along $n$ directions (canonical $n=4$, FCC $n=8$), e.g.,

$I_{k\ell} < \min\{I_{ij}, I_{mn}\}$

for each pixel $(i,j)$ and $(m,n)$ along direction $p$. Generalizations to arbitrary dimensions $d$ consider $2d + 2^d$ directions (Lacasa et al., 2017, Iacovacci et al., 2018).

Graph Features

Key descriptors include:

• Degree distribution $P(k)$—captures randomness, fractality, or periodicity.
• Local motifs ("visibility patches")—encode texture for classification (Iacovacci et al., 2018).
• Clustering coefficients, spectral measures, and average path lengths.

3. Polygonal, Terrain, and Rectangle Visibility Graphs

Polygon Visibility Graphs

For a polygon $P \subset \mathbb{R}^2$, the visibility graph connects two boundary vertices iff the segment joining them is entirely interior to $P$. Complexity of recognition is complete for the existential theory of the reals (ETR) when considering polygons with holes or internal/external visibility pairs (Boomari et al., 2018).

Terrain Visibility Graphs

Defined via $x$-monotone polygonal chains (terrains), terrain visibility graphs capture "above-terrain" relationships; they satisfy strong structural constraints (no large induced antiholes, strict order in cycles) and admit output-sensitive shortest path algorithms on terrain-like graphs, though recognition remains open (Froese et al., 2019).

Rectangle and Transparent Rectangle Visibility Graphs

• Rectangle visibility graphs (RVG): Vertices correspond to axis-parallel rectangles; edges indicate unobstructed horizontal or vertical lines of sight.
• Transparent rectangle visibility graphs (TRVG): Allow rectangles to be penetrable by sightlines—two rectangles are adjacent if any axis-aligned line meets both interiors, regardless of overlap by other rectangles. Families include all trees, cycles, threshold graphs, rectangular/triangular/hexagonal grids (Juntarapomdach et al., 17 Jun 2025).

4. Analytical Structure, Degree Distributions, and Enumeration

Degree Distributions

• Random time series/VG: $P(k) \sim C e^{-\alpha k}$; HVG yields closed-form shifted geometric distributions independent of marginal distributions: $P(k) = \frac{1}{n+1} (\frac{n}{n+1})^{k-n}$ for $k \geq n$ (Lacasa et al., 2017).
• Fractal/chaotic series: $P(k) \sim k^{-\gamma}$ with $\gamma$ tied to the fractal dimension or Hurst exponent (Lacasa et al., 2008).

Uniqueness and Enumeration of HVGs

• HVGs are outerplanar graphs with a Hamilton path (Gutin et al., 2010).
• Degree sequence uniquely determines an HVG for sequences without ties (Juhnke-Kubitzke et al., 2021); counted by Catalan numbers for distinct entries, and by large Schröder numbers if ties are allowed.

5. Computational Complexity and Recognition Problems

Point Visibility Graphs

Point visibility graphs (PVGs) associate vertices with planar points and edges with unobstructed segments. Recognition is $\exists\mathbb{R}$-complete: determining if a graph is a PVG is as hard as solving a system of real polynomial (in)equalities, due to arithmetic universality and combinatorial gadgets ("fan" construction) (Cardinal et al., 2015). Further, problems such as Feedback Vertex Set, Longest Induced Path, Bisection, and $\mathcal{F}$-free Vertex Deletion remain NP-hard, even on PVGs (Himmel et al., 2017).

Polygon Visibility Graphs

Recognition for visibility graphs of polygons with holes or internal/external graphs is $\exists\mathbb{R}$-complete (Boomari et al., 2018). Simple polygons (without holes) remain open.

Algorithmic Advances

• Efficient construction: Online/encoder–decoder algorithms using BST codecs enable $O(N\log N)$ HVG/NVG construction and real-time incremental updates for large time series (Yela et al., 2019).
• Output-sensitive shortest path: Specialized algorithms on terrain-like graphs achieve $O(d^*)$ complexity, where $d^*$ is the path-length (Froese et al., 2019).

6. Graph-Theoretic Properties: Connectivity, Thickness, Chromatic Number

The visibility structure imposes high connectivity: Non-collinear visibility graphs have diameter $2$, edge-connectivity equals minimum degree, and vertex-connectivity at least $(n-1)/(\ell-1)$ (where $\ell$ bounds collinearity), or at least $\frac{1}{2} \delta$ universally (Payne et al., 2011). For k-visibility graphs, thickness and chromatic numbers scale linearly with $k$, e.g., arc k-visibility graphs have thickness at most $3k+3$, chromatic number $6k+6$ (Babbitt et al., 2013, Sawhney et al., 2016).

7. Applications and Interpretational Insights

Visibility graphs bridge geometry and data-driven analysis in texture classification (Florindo et al., 2021), time-series characterization (Lacasa et al., 2008), image processing (Iacovacci et al., 2018), material science (Allen et al., 2018), turbulence discrimination (Lacasa et al., 2017), and architectural/urban design—where weather-induced attenuation is modeled via exponential decay of visibility weights (Schwartz et al., 2021). In neural architectures, constructing visibility graphs over convolutional feature maps injects fractal and periodic cues ignored by global pooling, improving classification without changing the network (Florindo et al., 2021). In image analytics, local motif histograms ("visibility patches") yield highly discriminative, robust features for texture recognition and compression (Iacovacci et al., 2018).

8. Controversies, Open Problems, and Research Horizons

• Graph-theoretic characterization and recognition for terrain and general polygon visibility graphs remain unresolved.
• Complexity separation: $\exists\mathbb{R}$-completeness often precludes efficient certifiability; for some cases, grid-realizability is likely undecidable.
• Independence of geometric invariants: Rectangle visibility graphs show that minimal area, perimeter, width, and height can require distinct representations (Caughman et al., 2022).
• Integrating physical phenomena: Weighted visibility graphs with attenuation coefficients enable urban analysis under variable weather (Schwartz et al., 2021).
• Class separation: Families of k-visibility graphs (bar vs arc vs semi-bar/semi-arc) are mutually incomparable; thresholds for extremal edge counts and chromatic numbers are tight up to constant factors.
• Enumeration: Catalan and Schröder numbers encode subtle combinatorial structure in HVG families (Juhnke-Kubitzke et al., 2021).

Visibility graphs remain central to translating geometric, temporal, and spatial structure into universal graph-homomorphic features, with continuous cross-fertilization between combinatorics, computational geometry, statistical physics, network science, and data-driven inference.