Topological Visibility: A Unified Perspective

• Topological visibility is a framework that redefines classical line-of-sight into relations based on closure, duality, and boundary structure.
• It unifies diverse settings—from Delaunay meshes and visibility graphs in time-series to complex domain geodesics and graph drawing—by encoding geometric connections as topological invariants.
• By translating local visibility rules into global topological properties, the theory offers actionable insights for analyzing proximity, connectivity, and homotopic structures across varied mathematical models.

Topological visibility denotes a family of constructions in which visibility is treated not merely as Euclidean line-of-sight but as a topological, proximity-theoretic, graph-theoretic, or boundary-geometric relation. In this literature, visibility can mean closure-based contact in a Delaunay mesh, dual adjacency between visibility graphs and merge trees, compact-core penetration of Kobayashi geodesics, or embedding-respecting visibility representations of nonplanar graphs and graphs on surfaces (Peters, 2015, Stephen, 2019, Bracci et al., 2021, Liotta et al., 2015). This suggests a unifying viewpoint: visibility becomes “topological” when it is encoded by closure, compactness, homology, duality, or admissible boundary structure rather than by metric straight-line obstruction alone.

1. Conceptual range and recurrent patterns

Across the cited literature, topological visibility appears in several distinct but structurally related forms.

Setting	Visibility object	Topological content
Delaunay meshes	Shared vertices or edges	Wallman proximity and local Leader uniform topology
Time-series visibility graphs	HVG, NVG, Cross-HV, horizon visibility	Clique complexes, persistent homology, merge-tree duality
Kobayashi geometry	Geodesics or almost-geodesics between boundary neighborhoods	Compact-core penetration, end compactification, boundary local connectivity
Graph drawing	Bar-visibility, L-visibility, toroidal/Klein-bottle visibility	Embedding constraints, right-angle crossings, surface identifications
Polygon spaces	Visibility graphs of polygons	Associahedral subcomplexes and deformation posets

A common pattern is the replacement of raw geometry by an invariant combinatorial or topological datum. In one direction, visibility becomes a proximity relation expressed by closure intersection. In another, visibility edges generate simplicial complexes whose Betti numbers, persistence diagrams, or merge trees summarize multiscale structure. In several complex variables, visibility is a condition on the behavior of geodesics near the boundary, closely tied to compactifications and local connectedness. In graph drawing, visibility is constrained by the topology of embeddings, by the torus or Klein bottle, or by controlled crossing structures (Adami et al., 2024, Masanta, 2024, Biedl, 2022).

This breadth is not terminological accident. A plausible implication is that “topological visibility” names a transfer principle: one starts with a visibility rule and then studies the induced topology, homotopy type, uniformity, or boundary compactification.

2. Proximity-theoretic visibility on Delaunay meshes

In proximal Delaunay geometry, visibility is defined on subsets of a Delaunay mesh built from a finite site set $S \subset \mathbb{R}^2$. A Delaunay edge $pq$ is characterized by Voronoï regions via

$$V_p \cap V_q = xy \Rightarrow pq \text{ is a Delaunay edge},$$

and a Delaunay triangle is a triangle all of whose edges are Delaunay edges. A Delaunay triangulation region $D$ is a collection of Delaunay triangles such that every pair of triangles is strongly near, meaning that any two triangles share a common edge. The paper defines visibility $v$ and strong visibility $\mathop{v}\limits^{\doublewedge}$ on subsets of such meshes and then identifies them with standard proximity-theoretic notions (Peters, 2015).

The basic visibility relation is

$$A \ v \ B \iff \exists \text{ triangle vertex } x \in \operatorname{cl}A \cap \operatorname{cl}B.$$

Strong visibility is

$A \ \mathop{v}\limits^{\doublewedge}\ B \iff \exists \text{ triangle edge } pq \text{ common to } A \text{ and } B.$

The crucial theorem is the equivalence

$A \delta B \iff A v B,$

where $\delta$ is Wallman proximity: $pq$0 Accordingly, the visibility relation $pq$1 is a Wallman proximity, and the strong visibility relation is also treated as a Wallman proximity (Peters, 2015).

This identification makes visibility a topological primitive. Closure can be recovered from visibility: $pq$2 The paper further states that a Delaunay triangulation region endowed with $pq$3 has a local Leader uniform topology. Concretely, for each $pq$4,

$pq$5

is the visibility-based neighborhood family used to generate the local uniform structure. In this framework, visibility is not simply adjacency in a mesh; it is the near relation of a proximity space, with associated closure, boundary, and local uniformity (Peters, 2015).

The same paper situates this notion against classical polygonal visibility, where two points are visible if the segment joining them stays inside the polygon. The contrast is explicit: classical visibility is unobstructed line-segment visibility, whereas Delaunay visibility is closure-based adjacency via shared vertices or edges. That shift from segment geometry to closure intersection is a central instance of topological visibility.

3. Visibility graphs, simplicial topology, and multiscale time-series structure

Visibility graphs convert scalar or multivariate data into graph topology. For a univariate series $pq$6, the horizontal visibility graph (HVG) connects $pq$7 when

$pq$8

For i.i.d. continuous random series, the HVG has the exact degree law

$pq$9

the long-distance visibility law

$$V_p \cap V_q = xy \Rightarrow pq \text{ is a Delaunay edge},$$0

and mean degree $$V_p \cap V_q = xy \Rightarrow pq \text{ is a Delaunay edge},$$1; the mean shortest path length scales logarithmically, giving a Small-World structure (Luque et al., 2010). In a related direction, the binary visibility graph for random Bernoulli sequences has the piecewise degree distribution

$$V_p \cap V_q = xy \Rightarrow pq \text{ is a Delaunay edge},$$2

and mean degree $$V_p \cap V_q = xy \Rightarrow pq \text{ is a Delaunay edge},$$3 (Ahadpour et al., 2010).

A major topological development is the horizon visibility graph $$V_p \cap V_q = xy \Rightarrow pq \text{ is a Delaunay edge},$$4, defined by augmenting a finite series $$V_p \cap V_q = xy \Rightarrow pq \text{ is a Delaunay edge},$$5 to

$$V_p \cap V_q = xy \Rightarrow pq \text{ is a Delaunay edge},$$6

Its decisive property is duality: $$V_p \cap V_q = xy \Rightarrow pq \text{ is a Delaunay edge},$$7 is exactly the dual of the merge tree of the weighted path associated to the time series, while the ordinary HVG is the weak dual obtained by removing the external regions. The persistence weighted horizon visibility graph is the metric dual, and applying the Elder Rule to the first Horton pruning recovers the barcode of the piecewise linear interpolation of the series (Stephen, 2019). This provides an explicit algebraic-topological foundation for visibility-graph methods.

Topological data analysis enters more directly in the study of natural visibility graphs (NVGs) of sandpile avalanche time series. For avalanche sizes $$V_p \cap V_q = xy \Rightarrow pq \text{ is a Delaunay edge},$$8, two nodes $$V_p \cap V_q = xy \Rightarrow pq \text{ is a Delaunay edge},$$9 are adjacent when

$D$0

The paper then equips edges with weights derived from local slopes and studies the clique complex of thresholded graphs. The resulting visibility graph of the Bak–Tang–Wiesenfeld model is reported to be scale free with degree exponent $D$1 and betweenness exponent $D$2. One-, two-, and three-dimensional simplex counts have power-law exponents $D$3, $D$4, and $D$5. Persistent entropy for $D$6 increases logarithmically with network size $D$7 (Adami et al., 2024). Here “topological visibility” explicitly means the topology of higher-order visibility relations: components, loops, voids, and their persistence across filtration scales.

The limited penetrable HVG family introduces a controlled relaxation of the horizontal visibility rule. In LPHVG$D$8, at most $D$9 intermediate points may be higher than both endpoints. For i.i.d. random series, the degree law is

$v$0

with mean degree

$v$1

Directed and image variants then support irreversibility tests and spatial discrimination of noise from chaos (Wang et al., 2017).

For multivariate series, the multilayer horizontal visibility graph (MHVG) uses ordinary HVG within each component and cross-horizontal visibility between lagged timestamps of different components. For rescaled components $v$2 and $v$3, cross-horizontal visibility is defined by

$v$4

This produces intra-layer, inter-layer, and all-layer topological measures, together with the novel ratio degree

$v$5

The paper reports that inter-layer edges preserve information about cross-dimension dependencies that would be lost in single-layer or multiplex mappings, but are not sufficient on their own; they complement the information carried by intra-layer edges (Silva et al., 2023).

A further branch of this literature studies topological properties of HVGs built from fractional Brownian motion. The clustering coefficient decreases with the Hurst index $v$6, mean shortest-path length increases exponentially with $v$7 for fixed $v$8, the graphs are fractal with box-counting dimension $v$9 decreasing with $\mathop{v}\limits^{\doublewedge}$0, and the networks remain assortative (Xie et al., 2010). This suggests that topological visibility can also serve as a coarse geometric encoding of persistence and roughness in stochastic processes.

4. Visibility in Kobayashi geometry and boundary topology

In several complex variables and hyperbolic complex geometry, visibility is a property of Kobayashi geodesics or almost-geodesics. For a bounded complete hyperbolic domain $\mathop{v}\limits^{\doublewedge}$1 and distinct boundary points $\mathop{v}\limits^{\doublewedge}$2, the pair $\mathop{v}\limits^{\doublewedge}$3 has visible geodesics if there exist disjoint neighborhoods $\mathop{v}\limits^{\doublewedge}$4 of $\mathop{v}\limits^{\doublewedge}$5, $\mathop{v}\limits^{\doublewedge}$6 of $\mathop{v}\limits^{\doublewedge}$7, and a compact set $\mathop{v}\limits^{\doublewedge}$8 such that every Kobayashi geodesic joining a point of $\mathop{v}\limits^{\doublewedge}$9 to a point of $$A \ v \ B \iff \exists \text{ triangle vertex } x \in \operatorname{cl}A \cap \operatorname{cl}B.$$0 intersects $$A \ v \ B \iff \exists \text{ triangle vertex } x \in \operatorname{cl}A \cap \operatorname{cl}B.$$1 (Bracci et al., 2021). This condition is equivalent to boundedness of the Gromov product at distinct Euclidean boundary points: $$A \ v \ B \iff \exists \text{ triangle vertex } x \in \operatorname{cl}A \cap \operatorname{cl}B.$$2 for a base point $$A \ v \ B \iff \exists \text{ triangle vertex } x \in \operatorname{cl}A \cap \operatorname{cl}B.$$3 (Bracci et al., 2021).

The same paper proves that every Gromov hyperbolic convex domain has the visibility property for any pair of distinct boundary points, and that Goldilocks domains and log-type domains also enjoy visibility. It further introduces localized criteria via $$A \ v \ B \iff \exists \text{ triangle vertex } x \in \operatorname{cl}A \cap \operatorname{cl}B.$$4-points and locally $$A \ v \ B \iff \exists \text{ triangle vertex } x \in \operatorname{cl}A \cap \operatorname{cl}B.$$5-strictly convex points. A bounded domain with Dini-smooth boundary in which every boundary point is locally $$A \ v \ B \iff \exists \text{ triangle vertex } x \in \operatorname{cl}A \cap \operatorname{cl}B.$$6-strictly convex has the visibility property (Bracci et al., 2021).

The 2024 extensions move from bounded domains in $$A \ v \ B \iff \exists \text{ triangle vertex } x \in \operatorname{cl}A \cap \operatorname{cl}B.$$7 to domains in arbitrary complex manifolds and explicitly accommodate non-complete Kobayashi metrics by working with $$A \ v \ B \iff \exists \text{ triangle vertex } x \in \operatorname{cl}A \cap \operatorname{cl}B.$$8-almost-geodesics. For a Kobayashi hyperbolic domain $$A \ v \ B \iff \exists \text{ triangle vertex } x \in \operatorname{cl}A \cap \operatorname{cl}B.$$9, a pair of distinct points satisfies visibility if every $A \ \mathop{v}\limits^{\doublewedge}\ B \iff \exists \text{ triangle edge } pq \text{ common to } A \text{ and } B.$0-almost-geodesic joining neighborhoods of the two points enters a fixed compact set; weak visibility restricts to $A \ \mathop{v}\limits^{\doublewedge}\ B \iff \exists \text{ triangle edge } pq \text{ common to } A \text{ and } B.$1. The paper introduces admissible compactifications $A \ \mathop{v}\limits^{\doublewedge}\ B \iff \exists \text{ triangle edge } pq \text{ common to } A \text{ and } B.$2, with totally disconnected ideal boundary $A \ \mathop{v}\limits^{\doublewedge}\ B \iff \exists \text{ triangle edge } pq \text{ common to } A \text{ and } B.$3, and proves that $A \ \mathop{v}\limits^{\doublewedge}\ B \iff \exists \text{ triangle edge } pq \text{ common to } A \text{ and } B.$4 is visible iff it is a visibility domain subordinate to any admissible compactification (Masanta, 2024).

A local-global theorem then states that $A \ \mathop{v}\limits^{\doublewedge}\ B \iff \exists \text{ triangle edge } pq \text{ common to } A \text{ and } B.$5 is visible iff it is locally visible and hyperbolically embedded in $A \ \mathop{v}\limits^{\doublewedge}\ B \iff \exists \text{ triangle edge } pq \text{ common to } A \text{ and } B.$6; the same equivalence holds for weak visibility (Masanta, 2024). Another theorem shows that if the set of non-visible points is totally disconnected, then the entire domain is visible. The paper also proves a Wolff–Denjoy-type theorem: for a taut, weakly visible domain, any holomorphic self-map either has relatively compact forward orbits or all forward iterates converge locally uniformly to a single ideal boundary point in an admissible compactification (Masanta, 2024).

In planar domains, visibility becomes closely tied to boundary topology. Totally disconnected subsets of the boundary are removable for visibility: if every pair of distinct points in $A \ \mathop{v}\limits^{\doublewedge}\ B \iff \exists \text{ triangle edge } pq \text{ common to } A \text{ and } B.$7 satisfies visibility and $A \ \mathop{v}\limits^{\doublewedge}\ B \iff \exists \text{ triangle edge } pq \text{ common to } A \text{ and } B.$8 is totally disconnected, then $A \ \mathop{v}\limits^{\doublewedge}\ B \iff \exists \text{ triangle edge } pq \text{ common to } A \text{ and } B.$9 is a visibility domain (Chandel et al., 2024). Under the Boundary Separation Property, a domain is a local (weak) visibility domain iff it is a global (weak) visibility domain (Chandel et al., 2024). For hyperbolic simply connected planar domains,

$A \delta B \iff A v B,$0

(Chandel et al., 2024). The same paper reformulates the Mandelbrot Local Connectivity conjecture in terms of visibility of the complement domain, and provides broad planar criteria for continuous or homeomorphic extension of conformal and biholomorphic maps up to the boundary.

This complex-analytic strand makes topological visibility precise in a different sense from visibility graphs. Here the visible object is the boundary geometry of the domain as seen by the Kobayashi metric, and the governing topological invariants are end compactness, local connectedness, prime-end behavior, and Gromov boundary identification.

5. Visibility representations in topological graph drawing

In graph drawing, visibility is a representation scheme: vertices are geometric objects, edges are unobstructed horizontal or vertical visibilities, and the topological constraints come from the embedding class of the graph. For IC-plane graphs, where each edge is crossed at most once and no two crossed edges share a vertex, every $A \delta B \iff A v B,$1-vertex IC-plane graph admits an L-visibility drawing in $A \delta B \iff A v B,$2 area computable in $A \delta B \iff A v B,$3 time (Liotta et al., 2015). In this model each vertex is an L-shape

$A \delta B \iff A v B,$4

each edge is a horizontal or vertical visibility segment, and crossings occur only between horizontal and vertical visibilities, hence at right angles. The same construction yields a RAC drawing with at most two bends per edge in $A \delta B \iff A v B,$5 area and $A \delta B \iff A v B,$6 time (Liotta et al., 2015).

The construction proceeds through augmentation to empty kites, removal of crossing edges to obtain a planar graph $A \delta B \iff A v B,$7, $A \delta B \iff A v B,$8-orientation of a contracted multigraph, reinsertion of selected edges to form a planar $A \delta B \iff A v B,$9, strong bar-visibility drawing of $\delta$0, and local conversion of bars into L-shapes so that the omitted crossing edges become horizontal visibilities. Topologically, the method translates IC-planarity into visibility constraints: each visibility is crossed at most once and no two crossed visibilities share an endpoint (Liotta et al., 2015).

On higher-genus surfaces, visibility representations remain possible but must respect the topology of the ambient surface. For a toroidal or Klein-bottle graph, vertices are horizontal segments and edges are vertical segments on a flat surface obtained from a rectangle by side identifications. The paper proves that every toroidal graph without loops has a visibility representation on the rectangular flat torus, and every graph without loops embedded on the Klein bottle has a visibility representation on the rectangular flat Klein bottle (Biedl, 2022). The core construction reduces the problem to a planar visibility representation on a flat cylinder, builds non-crossing $\delta$1-$\delta$2 path systems, enforces a path-respecting bipolar orientation, and then reinstates wrap-around edges along exclusive columns. For the Klein bottle, additional symmetry of columns is required because top and bottom are identified in opposite directions (Biedl, 2022).

In this literature, topological visibility is neither proximity nor boundary behavior. It is the problem of realizing a graph or surface embedding by visibilities while preserving global topological structure: independent crossings, homotopy of wrap-around edges, and non-orientable identifications.

6. Visibility graphs of polygons and associahedral deformation spaces

For simple polygons, visibility graphs determine which diagonals are geometrically admissible, and this in turn controls an associahedron-like polytopal complex. Given a simple polygon $\delta$3, its visibility graph $\delta$4 has the polygon’s vertices as vertices and edges $\delta$5 whenever the segment $\delta$6 is either a boundary edge or an interior diagonal contained in $\delta$7. Let $\delta$8 be the poset of convex diagonalizations of $\delta$9, ordered by adding diagonals. The paper constructs a polytopal complex $pq$00 whose face poset is isomorphic to $pq$01, and proves that $pq$02 is a subcomplex of the classical associahedron $pq$03 for an $pq$04-gon (0903.2848).

If $pq$05 is the minimum number of diagonals required to convexly diagonalize $pq$06, then

$pq$07

If a set of noncrossing diagonals $pq$08 divides $pq$09 into polygons $pq$10, then

$pq$11

and if each $pq$12 is convex with $pq$13 sides, this becomes

$pq$14

The complex $pq$15 is connected and, more strongly, contractible (0903.2848).

The same paper introduces a deformation space of polygons organized by visibility graphs. Two polygons are $pq$16-equivalent if they have the same visibility graph, and $pq$17-isotopic if they can be continuously deformed through polygons with fixed visibility graph. The set of $pq$18-isotopy classes forms a deformation poset $pq$19, ordered by one-edge changes in visibility graphs under controlled deformations. The maximal element is the convex polygon; minimal elements are polygons with unique triangulations. This organizes families of subcomplexes $pq$20 inside $pq$21 by visibility loss or gain (0903.2848).

This polygonal setting is one of the clearest literal realizations of topological visibility. A purely geometric notion—who sees whom inside a polygon—determines a contractible cell complex, a deformation poset of isotopy classes, and a concrete substructure of the associahedron.

7. Visibility manifolds and volume-controlled topological complexity

In Riemannian geometry, a Hadamard manifold $pq$22 satisfies the visibility axiom if for every pair of distinct points $pq$23, there exists a geodesic $pq$24 with

$pq$25

A complete finite-volume manifold $pq$26 is then a visibility manifold when its universal cover has this property (Senska, 2019). The paper considers the case

$pq$27

with no pinching away from $pq$28, so curvature may approach zero while visibility still holds.

The principal result is an efficient simplicial model for the thick part $pq$29: there exist constants $pq$30 and $pq$31, depending only on dimension, such that the pair $pq$32 is homotopy equivalent to a simplicial pair $pq$33 with at most $pq$34 vertices and vertex degree bounded by $pq$35 (Senska, 2019). From this, one obtains linear volume bounds for Betti numbers,

$pq$36

for some $pq$37, all $pq$38, and every coefficient field $pq$39, and torsion bounds

$pq$40

for some $pq$41, all $pq$42, except the case $pq$43 (Senska, 2019).

The thick–thin decomposition is likewise controlled: the thick part is compact with boundary, the number of thin components is bounded linearly in volume, and each thin component is either a tube, homeomorphic to a $pq$44-bundle over $pq$45, or a cusp, homeomorphic to $pq$46 with strong deformation retraction onto its boundary (Senska, 2019). Visibility is used to classify stabilizers of thin components as parabolic or hyperbolic with common fixed data and to construct geodesic flows retracting the thick part onto a shrunken core.

This setting pushes topological visibility to a genuinely global scale. Visibility at infinity constrains the topology of the manifold so strongly that both free and torsion homology are controlled by volume. A plausible interpretation is that visibility serves here as a large-scale substitute for pinched negative curvature: it is weak enough to allow curvature approaching $pq$47, but still strong enough to suppress uncontrolled topological complexity.

Topological visibility is therefore not a single doctrine but a recurrent structural move. Whether in Delaunay meshes, time-series complexes, Kobayashi domains, visibility drawings, polygon spaces, or negatively curved manifolds, the visible relation is promoted to a topological object—proximity, dual complex, admissible boundary relation, embedding constraint, or volume-controlled homotopy model—and then studied through the invariants of topology rather than by line-of-sight geometry alone (Peters, 2015, Adami et al., 2024, Chandel et al., 2024, Senska, 2019).