Visibility-Aware Graph Models

• Visibility-Aware Graph Models are frameworks where edges form based on geometric, combinatorial, or perceptual visibility criteria, enabling nuanced network connectivity.
• They integrate diverse algorithmic approaches for pursuits, time series, and urban analytics, demonstrating robust performance in spatial and image processing tasks.
• The models exhibit distinctive properties such as planarity, chromatic bounds, and mutual-visibility metrics, offering key insights for both theoretical investigations and real-world applications.

Visibility-aware graph models are mathematical frameworks and algorithmic paradigms in which visibility—defined via geometric, combinatorial, or information-theoretic criteria—determines the edge structure in a graph. Such models emerge in contexts ranging from pursuit-evasion and geometric network design, to time series analysis, image processing, and urban visibility analytics. Visibility relations can be determined by occlusion, limited field of view, capacity to traverse $k$ obstacles, proximity range, or contextually derived perceptual cues. These models support rich structural theory, computational algorithms, and a spectrum of applications.

1. Formal Models of Visibility-Aware Graphs

Visibility-aware graphs arise whenever vertices/entities are connected if and only if they are mutually "visible" according to model-specific criteria. Principal classes include:

• Geometric Visibility Graphs: Vertices represent geometric objects (points, segments, polygons), and edges connect pairs not mutually occluded by obstacles. Classical variants include polygon visibility graphs, terrain visibility graphs, and unit-disk visibility graphs, where limited sensing range truncates visibilities (Çağırıcı et al., 2021, Froese et al., 2019).
• Limited Visibility Graphs (LVG): Visibility is not absolute but constrained—e.g., each agent at node $x$ observes only a subset defined by a visibility matrix $B$, possibly derived from graph distance or embedding (Kehagias et al., 2021).
• $k$-Visibility Graphs: Edges are present if two objects can "see" each other through at most $k$ obstacles. This interpolates between planar (no crossings, $k=0$) and highly nonplanar ($k\gg1$) connectivity regimes (Babbitt et al., 2013).
• Time Series and Image Visibility Graphs: Data points or pixels are nodes; edges encode geometric or sequential local visibility, e.g., through intervals in a scalar sequence or along rows, columns, and diagonals in an image (Iacovacci et al., 2015, Iacovacci et al., 2018).
• Perceptual and Observation-based Visibility: Graphs encoding the detectability of objects in complex environments, often integrating machine perception and contextual information (Fan et al., 17 May 2025).

These definitions are unified by the use of visibility constraints—occlusion, field-of-view, range, or environmental context—to generate adjacency.

2. Structural Properties and Extremal Results

Visibility-aware graph models display distinctive extremal and structural characteristics shaped by their construction rules:

• Planarity and Thickness: Classical bar-visibility ($k=0$) graphs are planar and bipartite. When obstacle tolerance parameter $k$ increases, graph thickness (minimal number of planar subgraphs needed to cover all edges) grows, bounded as $\lceil \tfrac23 (k+1)\rceil \leq \Theta(G) \leq 2k$ for semi-bar $k$-visibility graphs (Babbitt et al., 2013).
• Edge and Chromatic Number Bounds: $k$-visibility frameworks yield sharp upper bounds for edge densities and chromatic numbers, with arc/circle $k$-visibility graphs on $n$ vertices achieving at most $(k+1)(3n - k - 2)$ edges for $n > 4k+4$, and chromatic number at most $6k+6$. When $k=0$, maximal graphs coincide with known planar graph bounds (Babbitt et al., 2013).
• Terrain Visibility Graphs (TVGs): TVGs correspond to $x$-monotone terrains with edges between pairs $(p_i, p_j)$ such that no intermediate terrain vertex obstructs the line of sight. All TVGs are persistent (satisfying X- and Bar-properties), and, notably, cannot contain antiholes of size $k\geq6$ (Froese et al., 2019).
• Mutual-Visibility Parameters: For a graph $G$ and subset $S\subseteq V(G)$, key parameters include the mutual-, total-, outer-, and dual-mutual-visibility numbers ($\mu(G), \mu_t(G), \mu_o(G), \mu_D(G)$). These measure the size of the largest subset in which every vertex pair is mutually visible in the shortest-path sense (no internal vertices of the path in $S$). There exist strong connections to extremal functions such as the Zarankiewicz number and the Turán graph edge counts (Cicerone et al., 4 Jan 2024).

3. Algorithms and Computational Complexity

Algorithmic developments in visibility-aware graphs address both model-specific constructions and computational challenges:

• Graph Search with Limited Visibility: The LVGS (Limited Visibility Graph Search) algorithm computes optimal pursuit schedules on graphs with arbitrary visibility matrices by reducing state space via 'information-state' Markov abstraction, encoding positions and contamination status of invisible components (Kehagias et al., 2021). The information-state digraph $G_I$ supports Dijkstra/BFS search for minimal clearance time but suffers from combinatorial explosion in $|V_I|$.
• $k$-Visibility and Chromatic/Coloring Complexity: While unit-disk intersection graphs support polynomial clique detection, NP-completeness arises for graph coloring on more general visibility-aware graphs. For instance, 3-colorability is NP-complete in unit-disk visibility graphs with segments or with polygons containing holes (Çağırıcı et al., 2021). In $k$-visibility, chromatic and thickness constraints scale as functions of $k$.
• Time Series and Image Algorithms: Efficient algorithms exist for constructing both 1D and 2D visibility graphs. Sequential/online algorithms for time series (e.g., the LOT framework) attain $O(N)$ amortized time per window for both Natural and Horizontal Visibility Graphs, enabling real-time analytics (Huang et al., 2023). Image Visibility Graphs admit $O(N^2)$ construction for $N \times N$ images (Iacovacci et al., 2018).
• Shortest Path and Dominating Set in Terrains: Terrain visibility graphs admit tailored shortest-path algorithms exploiting the vertex order and the X-property, achieving $O(d^*)$ time for distance $d^*$ queries after preprocessing, and polynomial-time dominating set computation for 'funnel' terrains (Froese et al., 2019).
• Mutual-Visibility in General Graphs: Approximation algorithms, based on reductions to uniform hypergraph independent set, yield $\Omega(\sqrt{n/\overline{D}})$-size mutual-visibility sets, with $\overline{D}$ the average distance in $G$. Strong inapproximability results hold for all variants of mutual-visibility—APX-hardness and $n^{1/3-\varepsilon}$ lower bounds for general graphs (Bilò et al., 29 Jun 2024).

4. Applications Across Domains

Visibility-aware graph models underpin diverse applications:

• Robotics and Surveillance: LVGS, unit-disk, and $k$-visibility frameworks model sensor and agent networks, visibility-based pursuit-evasion, and intruder detection in discretized and polygonal environments (Kehagias et al., 2021, Çağırıcı et al., 2021).
• Networked Sensing/Urban Analytics: Heterogeneous visibility graphs with observer and landmark nodes, often integrating perception via vision-LLMs, support urban landmark visibility analysis more aligned to human perception than geometric LoS, opening new research avenues in urban planning and social-heritage networks (Fan et al., 17 May 2025).
• Time Series Analysis: Visibility graphs extract and encode the dynamics of time series (e.g., stochastic vs. chaotic), with motif profiles providing highly robust, low-dimensional feature sets for time series classification, trend detection, and unsupervised learning (Iacovacci et al., 2015).
• Image Processing and Compression: Image visibility graphs and derived features such as visibility patches enable computationally efficient image filtering, denoising, compression via degree-thresholding, and high-accuracy image classification across material, biomedical, and texture domains (Iacovacci et al., 2018).
• Combinatorial Optimization and Extremal Problems: Calculation of mutual-visibility numbers in graph products, extremal constructions in $k$-visibility, and the evaluation of associated parameters in cographs and line graphs enrich the toolkit of extremal combinatorics and inform graph-theoretic studies linked to visibility (Cicerone et al., 4 Jan 2024).

5. Open Problems and Theoretical Directions

Visibility-aware modeling interfaces with several open questions and active research areas:

• State-space Reduction and Scalability: Achieving tractable information-state representations for multi-agent or high-order LVGS, e.g., via kernelization or parameterization by treewidth, remains challenging (Kehagias et al., 2021).
• Recognition and Characterization: Full graph-theoretic characterizations for terrain and polygonal visibility graphs (including forbidden subgraph hierarchies) are incomplete, with persistent but non-visibility graphs known (Froese et al., 2019).
• Algorithmic Hardness: The status of NP-completeness for unit-disk visibility graphs under additional constraints (e.g., unit-length, axis-aligned segments), clique detection complexities for UDVGs, and the impact of range-limitation on classic visibility problems are unresolved (Çağırıcı et al., 2021).
• Integration of Perceptual and Contextual Visibility: Heterogeneous graphs integrating image-based detection and context-aware edge definitions suggest a new paradigm for data-driven urban analytics and network design, raising questions on robustness, generalizability, and the mathematical implications of incorporating perceptual semantics (Fan et al., 17 May 2025).
• High-dimensional and Multivariate Extensions: The extension of planar visibility graph ideas to multidimensional time series and volumetric data, including analytical paper of motifs and features in higher dimensions, is at an early stage (Liu et al., 2014, Iacovacci et al., 2018).

6. Connections to Extremal and Probabilistic Graph Theory

Visibility-aware models regularly invoke and extend extremal results:

• Mutual-visibility numbers in Cartesian and tensor products of complete graphs solve Zarankiewicz-type problems, and in line graphs, Turán-type bounds appear (Cicerone et al., 4 Jan 2024).
• $k$-visibility transitions the combinatorial structure from strictly planar ($k=0$) through dense/high-chromatic nonplanar regimes.
• Degree distributions, clustering, and motif frequency profiles in visibility graphs serve as statistical invariants distinguishing random, periodic, and chaotic series (Iacovacci et al., 2015, Liu et al., 2014).
• The relationship between mutual-visibility and general position numbers in graphs has distinct inapproximability profiles, exposing subtleties in their combinatorial hierarchies (Bilò et al., 29 Jun 2024).

---

Visibility-aware graph models constitute an intersection of geometry, combinatorics, algorithmic design, and emerging perceptual/computational paradigms. They facilitate rigorous analyses and efficient computation in settings where visibility, broadly construed, shapes connectivity—enabling new approaches to pursuit-evasion, pattern recognition, network design, and contextual analytics in real and virtual environments.