Graph of Regions: Concepts and Constructions

• Graph of Regions is a set of constructions that compress higher-dimensional or complex regional data into graphs reflecting adjacency, intersection, or connectivity.
• It encompasses intersection graphs, quotient graphs from level-set collapsing, and boundary graphs that yield structural separators and spectral insights.
• Applied forms model real-world systems in epidemiology, image analysis, and dynamical systems by representing spatial or latent regions with connectivity graphs.

“Graph of Regions” is a polysemous technical term rather than a single standardized object. In current literature it denotes, among other things, an intersection graph built from connected regions of a base graph, a quotient graph obtained by collapsing connected components of level sets of a projection of a geometric region, the graph of an extremal boundary function for feasible homomorphism-density regions, and application-specific networks whose vertices are spatial, image, or latent regions (Lee, 2016, Kitazawa, 12 Mar 2025, Hatami et al., 2016, Darapu et al., 2022). What unifies these usages is that a higher-dimensional or more structured “region” space is compressed into a graph encoding adjacency, intersection, reachability, or critical change.

1. Terminological scope

The literature uses the phrase in several non-equivalent ways. The underlying constructions differ by what is collapsed or related.

Usage	Basic construction	Representative papers
Region intersection graph	vertices are connected regions; edges record nonempty intersection	(Lee, 2016)
Poincaré–Reeb graph of a region	connected components of level sets of a projection are collapsed to points	(Kitazawa, 29 Jan 2025, Kitazawa, 12 Mar 2025)
Boundary graph of a feasible region	one studies the graph of an extremal boundary function of a density region	(Hatami et al., 2016)
Applied graph of regions	regions are geographic, image, or latent cells; edges encode mobility, adjacency, or reachability	(Darapu et al., 2022, Manzo, 2019, Aladin et al., 26 Feb 2026)

A recurrent source of ambiguity is that some papers use “graph of regions” for a genuine graph-theoretic object built from regions themselves, while others use it for a graph extracted from a region by quotienting level sets, and still others for the graph of a boundary curve of a feasible set. The distinction is structural: intersection graphs are combinatorial from the outset, Poincaré–Reeb graphs are quotient spaces, and homomorphism-density “graphs of regions” are boundary parametrizations rather than finite graphs.

2. Region intersection graphs in structural graph theory

In structural graph theory, the standard formalization is the region intersection graph over a base graph $G_0=(V_0,E_0)$. A graph $G=(V,E)$ is a rig over $G_0$ if there exists a family of connected subsets

$\{R_u\subseteq V_0: u\in V\}$

such that, for distinct $u,v\in V$,

$\{u,v\}\in E \iff R_u\cap R_v\neq\emptyset.$

Vertices of $G$ therefore represent connected regions of the base graph, and adjacency records intersection (Lee, 2016).

This definition subsumes several classical graph classes. Interval graphs arise when $G_0$ is a path and each region is a connected subpath. Finite string graphs are exactly the rigs over planar graphs: every string graph is a rig over some planar graph, and every finite rig over a planar graph is a string graph (Lee, 2016). The base graph need not resemble the intersection graph itself; indeed, a rig over a sparse base may be dense and may contain large cliques.

The main structural theorem is a separator result. If $G_0$ excludes $K_h$ as a minor and $G=(V,E)$0 has $G=(V,E)$1 edges, then $G=(V,E)$2 has a $G=(V,E)$3-balanced separator of size at most

$G=(V,E)$4

with $G=(V,E)$5 depending only on $G=(V,E)$6, and the paper gives $G=(V,E)$7 (Lee, 2016). For planar $G=(V,E)$8, this yields the optimal $G=(V,E)$9 separator bound for string graphs, confirming the Fox–Pach conjecture and improving the earlier $G_0$0 bound.

The proof does not rely on the rig itself being minor-free. Instead it combines conformal graph metrics, extremal spread, observable spread, LP duality with multiccommodity flows, and padded random partitions derived from the absence of careful $G_0$1-minors in rigs over $G_0$2-minor-free bases. Under bounded maximum degree, the same framework yields spectral control: $G_0$3 so recursive spectral partitioning finds balanced separators algorithmically (Lee, 2016). In this usage, a graph of regions is an intersection graph whose hidden structure is inherited from the base graph.

3. Poincaré–Reeb graphs of algebraic regions

A different tradition, originating in recent work on real algebraic domains, uses “graph of a region” for the quotient obtained by collapsing connected components of level sets of a projection. In the plane, one starts with a bounded connected region $G_0$4 surrounded by finitely many non-singular real algebraic curves. In the refined setting, these curves may intersect, but only with normal crossings: at most two curves meet at a point in $G_0$5, and their tangent lines span the tangent space of the plane (Kitazawa, 29 Jan 2025).

Fix the canonical projection

$G_0$6

Two points of $G_0$7 are identified when they lie in the same connected component of a common level set of $G_0$8. The quotient is the Poincaré–Reeb graph. Its vertices are the quotient points whose preimages contain either a boundary intersection point or a critical point of the projection along a smooth boundary arc; edges are the remaining one-dimensional pieces. The projection descends to a real-valued function on the graph, so the result is naturally a V-digraph in the sense of a graph equipped with a height function inducing edge orientations (Kitazawa, 29 Jan 2025, Kitazawa, 12 Mar 2025).

This framework generalizes the earlier setting of disjoint boundary curves. For refined algebraic domains, the finite set

$G_0$9

collects the relevant boundary intersections and projection-critical points, and the larger characteristic set

$\{R_u\subseteq V_0: u\in V\}$0

is treated as part of the data of a refined algebraic domain with poles (Kitazawa, 12 Mar 2025). These finite sets determine the vertex set of the Poincaré–Reeb graph and constrain admissible geometric modifications.

A realization theorem shows that if a graph $\{R_u\subseteq V_0: u\in V\}$1 carries a piecewise smooth function $\{R_u\subseteq V_0: u\in V\}$2 that is injective on each edge, arises as $\{R_u\subseteq V_0: u\in V\}$3 for some piecewise smooth embedding $\{R_u\subseteq V_0: u\in V\}$4, has no degree-2 vertices, and attains local extrema only at degree-1 vertices, then there exists a refined algebraic domain whose Poincaré–Reeb V-digraph is weakly isomorphic to $\{R_u\subseteq V_0: u\in V\}$5 (Kitazawa, 29 Jan 2025). In this usage, a graph of regions is a quotient graph encoding how the connected components of vertical slices are born, merge, and disappear.

4. Explicit realizations by circles, parabolas, and cylinders

A substantial body of recent work studies explicit families of regions whose Poincaré–Reeb graphs realize prescribed trees. For planar regions bounded by parabolas, every finite tree is realizable: each tree is isomorphic to the Poincaré–Reeb graph of some RA-region surrounded by parabolas congruent to one fixed parabola or to another suitably chosen parabola (Kitazawa, 5 Feb 2026). The construction uses the projection $\{R_u\subseteq V_0: u\in V\}$6, identifies vertices with singular fibers, and then builds arbitrary branching by arranging horizontal or vertical parabolas and locally refining edges by adding further parabolas.

For regions surrounded by circles, the analogous object is an SS-region. Its Poincaré–Reeb graph is formed from connected components of the slices

$\{R_u\subseteq V_0: u\in V\}$7

The paper develops two local circle-addition moves, MBCC addition and SSCC addition, and analyzes their graph effect: MBCC addition inserts two vertices into an edge and adds a pendant edge, while SSCC addition inserts two vertices into an edge without adding a new leaf (Kitazawa, 9 Nov 2025). The resulting family of trees is characterized inductively, with explicit parity constraints on how vertices may be inserted along edges.

A further refinement equips Poincaré–Reeb graphs of circle arrangements with labels. In a normally inductive arrangement of circles, each circle is subdivided by eight $\{R_u\subseteq V_0: u\in V\}$8-poles into $\{R_u\subseteq V_0: u\in V\}$9-arcs, and labels on edges and vertices record which arcs and which circles appear in the preimages of graph elements under the quotient map (Kitazawa, 21 Feb 2025). These labels encode not just the abstract graph but its placement relative to the circle geometry.

Higher-dimensional analogues replace circles by cylinders of circles. In $u,v\in V$0, an RA-region can be formed by intersecting cylinders of disks and complements of disk-cylinders, then projected by

$u,v\in V$1

The resulting Poincaré–Reeb V-digraph realizes broad families of rooted trees: balanced trees with an extra incoming edge at the root, and more generally trees obtained by gluing two balanced trees with opposite orientations at their roots (Kitazawa, 17 Dec 2025). This makes the graph-of-regions viewpoint explicitly higher-dimensional and algebraic.

These realizability results suggest a sharp distinction from region intersection graphs. Here the graph is not built from pairwise intersection of regions; it is produced by collapsing fiber components of a projection, and the main question is which graphs can arise as quotient skeletons of explicitly defined algebraic regions.

5. Feasible regions of homomorphism densities

In extremal graph theory, “graph of the region” can mean the graph of an extremal boundary function of a feasible set of homomorphism densities. For a finite graph $u,v\in V$2 and a finite graph $u,v\in V$3, the induced homomorphism density is

$u,v\in V$4

where $u,v\in V$5 counts strong homomorphisms preserving adjacency and non-adjacency. For a finite list $u,v\in V$6, the feasible region

$u,v\in V$7

is the closure of all density vectors $u,v\in V$8, equivalently the set of graphon density vectors $u,v\in V$9 (Hatami et al., 2016).

The classical case $\{u,v\}\in E \iff R_u\cap R_v\neq\emptyset.$0 yields the edge–triangle region. Its boundary is exceptionally regular: Kruskal–Katona and Razborov’s flag algebra analysis imply that the boundary is a countable union of algebraic curves and hence differentiable almost everywhere. For the Turán graphon $\{u,v\}\in E \iff R_u\cap R_v\neq\emptyset.$1,

$\{u,v\}\in E \iff R_u\cap R_v\neq\emptyset.$2

and boundary pieces arise from Turán-type graphons and their perturbations (Hatami et al., 2016).

Hatami and Norin show that this regularity fails dramatically in general. There exists a quantum graph $\{u,v\}\in E \iff R_u\cap R_v\neq\emptyset.$3 such that the function

$\{u,v\}\in E \iff R_u\cap R_v\neq\emptyset.$4

where the minimum is taken over graphons satisfying

$\{u,v\}\in E \iff R_u\cap R_v\neq\emptyset.$5

is nowhere differentiable on some interval $\{u,v\}\in E \iff R_u\cap R_v\neq\emptyset.$6 (Hatami et al., 2016). In fact,

$\{u,v\}\in E \iff R_u\cap R_v\neq\emptyset.$7

throughout that interval. The construction uses finitely forcible families of infinite lexicographic products, so the boundary curve becomes a self-similar encoding of binary choices across infinitely many scales. In this meaning, the “graph of regions” is not a finite graph but a boundary graph of a feasible density region, and the main theme is regularity versus pathology.

6. Applied graphs of regions

Applied literature uses the term in yet other concrete senses. In epidemiological network modeling, a graph of regions is a weighted undirected graph

$\{u,v\}\in E \iff R_u\cap R_v\neq\emptyset.$8

whose nodes are geographic regions and whose edge weights encode mobility intensity (Darapu et al., 2022). An SIR meta-population model runs on this graph, and the infection fraction $\{u,v\}\in E \iff R_u\cap R_v\neq\emptyset.$9 is treated as a graph signal. Two local signal-variation metrics are then defined: $G$0 and

$G$1

These identify influential regions whose infection level is both spatially anomalous and, for the temporal metric, rapidly changing (Darapu et al., 2022).

In image analysis, the Attributed Relational SIFT-based Regions Graph (ARSRG) is a hierarchical graph of regions built from a region adjacency graph together with SIFT descriptors. Region nodes represent segmented image regions, region–region edges encode adjacency, region–SIFT edges attach keypoints to their containing regions, and optional SIFT–SIFT edges encode local proximity inside a region (Manzo, 2019). The construction supports graph matching, graph embedding, bag-of-graph-words models, and kernel methods, with reported gains when segmentation is meaningful.

In dynamical-systems learning, V-MORALS constructs a graph of regions in latent space. A learned latent space $G$2 is partitioned into cells; each cell is a region, and a directed edge $G$3 is added when a Lipschitz-inflated reachable set from $G$4 intersects $G$5 (Aladin et al., 26 Feb 2026). Strongly connected components of this cell graph are then collapsed to produce a Morse Graph, whose nodes are recurrent regions and whose leaves represent attractors. Regions of attraction are computed as sets of cells with directed paths to an attractor SCC. This is a graph of regions in the literal discretized-state-space sense.

These applied uses share a common pattern: the nodes are regions defined by domain knowledge—geographic, segmented, or latent—and graph structure is then used to quantify influence, similarity, or reachability.

7. Other region-based graph formalisms

Several neighboring formalisms are technically distinct but conceptually related. In hyperplane-arrangement theory, a bigraphical arrangement associates two hyperplanes to each edge of a graph $G$6, and one studies the regions of the complement. Those regions admit Pak–Stanley labels, and for any admissible parameter list the label set is exactly the set of parking functions of the augmented graph $G$7; in the generic case the number of regions is

$G$8

while for $G$9 the Shi arrangement achieves the minimum $G_0$0 regions (Hopkins et al., 2012). Here the natural “graph of regions” is the adjacency graph of arrangement chambers, although that terminology is not the paper’s own.

In approximate inference, a region graph in generalized belief propagation is a DAG whose nodes are clusters of variables and factors, with edges representing parent–child subregion relations. The region-based free energy

$G_0$1

is invariant under three operations—split, merge, and death—under stated conditions, which motivates the notion of weakly irreducible regions and the region pursuit algorithm for adding new regions under a region-width complexity bound (Welling, 2012). This is not a geometric graph of regions but a graph whose nodes are inference regions.

For compact regions in $G_0$2, the Blum medial axis supplies yet another graph-of-regions paradigm. Damon and Thomas decompose the medial axis into irreducible medial components and encode the result by a two-level extended graph structure: a top-level graph records how irreducible components attach, and for each component a second-level graph records how medial sheets with boundary attach to 4-valent Y-network graphs (Damon, 2009). Because the medial axis is a strong deformation retract of the region, this graph-of-graphs structure computes homology and the fundamental group and characterizes contractible regions.

Taken together, these literatures show that “graph of regions” is best understood as a family of constructions rather than a single definition. The term may denote an intersection graph, a quotient graph, a chamber adjacency graph, a cluster DAG, or a multilevel medial skeleton. The shared theme is that a complicated regional geometry or combinatorics is rendered tractable by passage to a graph, but the mathematical content of that passage depends entirely on what the regions are and which relation—intersection, level-set connectivity, adjacency, or message passing—defines the edges.