Gabriel Edited Set: Graphs & Quotient Rings

• Gabriel Edited Set is a concept encompassing both witness Gabriel graphs, which edit edge structures in computational geometry, and Gabriel quotient rings, which localize noncommutative rings using auxiliary sets.
• Witness Gabriel graphs control graph connectivity by removing edges based on point exclusions, ensuring planarity and allowing edge counts from 0 to complete connectivity.
• Gabriel quotient rings generalize classical localization by using Gabriel filters to yield intermediate rings with controlled torsion properties in noncommutative settings.

The Gabriel Edited Set refers to two sophisticated algebraic and geometric constructions: witness Gabriel graphs in computational geometry and Gabriel quotient rings in noncommutative ring theory. Both rely on the principle of "editing" an ambient structure—either a proximity graph or a localization of a ring—according to constraints imposed by an auxiliary set, termed witness set or Gabriel filter. These constructs enable fine-grained control over graph structure in the plane and localizations in module theory, respectively, expanding the toolbox for analysis in discrete geometry and algebra.

1. Witness Gabriel Graphs: Definitions and Fundamental Properties

Let $P$ and $W$ be finite point sets in general position in the Euclidean plane. For $a, b \in P$, define the closed disk with diameter $ab$ as

$$D(a,b) = \{\, x \in \mathbb{R}^2 : \|x-(a+b)/2\| \le \|a-b\|/2 \, \}.$$

The witness Gabriel graph $\GG^{-}(P,W)$ is the geometric graph with vertex set $P$ and edge set

$$E=\left\{\{a,b\}\subset P : D(a,b)\cap (W\setminus\{a,b\}) = \emptyset \right\}.$$

An edge $ab$ exists if and only if the closed diametral disk for $ab$ contains no point of $W$ other than possibly $a$ and $b$ themselves. Key special cases:

• The classical Gabriel graph arises when $W=P$, i.e., $\GG(P)=\GG^{-}(P,P)$.
• For $W = \emptyset$, the resulting graph is the complete graph on $P$, $\GG^{-}(P,\emptyset) = K_{|P|}$.

Adding points to $W$ can only remove edges from $\GG^{-}(P,W)$; thus, $W$ acts as an "editor" on the edge set, forbidding pairs whose diametral disks are stabbed by a witness (Aronov et al., 2010).

Structural properties include:

• Planarity: By inclusion $\GG^{-}(P,W)\subseteq\GG(P)$ and the planarity of the standard Gabriel graph, every witness Gabriel graph is planar.
• Edge-count Variability: For $|P|=n$, by suitable choice of $W$, the number of edges in $\GG^{-}(P,W)$ attains every integer from 0 to $\binom{n}{2}$.
• Edge Stabbing Capacity: It is always possible to remove all edges by selecting $n-1$ witnesses, but some point configurations require at least $\frac{3}{4}n - o(n)$ witnesses to achieve this.

2. Verification and Construction Algorithms

Algorithms for constructing or verifying witness Gabriel graphs operate via geometric arrangements and Voronoi diagrams:

• Half-plane intersection algorithm: For each $p\in P$ and $q\in W$, draw the perpendicular bisector to $pq$ and define the closed half-plane containing $p$. The intersection $I_p$ of all such half-planes identifies the region where a neighbor $r\in P$ must lie for $pr$ to be an edge. Total time complexity: $O(n^2)$ for $n=|P|+|W|$.
• Voronoi-based algorithm: The Voronoi diagram of $W$ is constructed in $O(n\log n)$ time. For each $p\in P$ and each candidate edge $pr$, the edge is present iff the midpoint of $pr$ lies in the Voronoi cell of $p$. Again, total complexity is $O(n^2)$.
• Output-sensitive approach: The witness Delaunay graph $\DG^{-}(P,W)$ can be built in $O(e\log n + n\log^2 n)$ where $e=|\DG^{-}(P,W)|$, and edge-disk emptiness is testable in $O(\log n)$ per edge.

Given a straight-line embedding of a graph $G=(V,E)$, there is an $O(|V|^2\log|E|)$ algorithm to decide if a witness set $W$ exists so that $G = \GG^{-}(V,W)$ and to produce such a $W$ (Aronov et al., 2010).

3. Characterization, Drawability, and Structural Boundaries

Not all graphs are witness Gabriel; however, explicit characterizations exist for significant families:

• Every tree admits a witness Gabriel drawing. This is achieved via classic geometric decomposition: rooting the tree, laying out child edges in shrinking cones, and inserting witnesses to obstruct non-tree edges without interfering with tree structure.
• All complete bipartite graphs $K_{m,n}$ can be realized by placing each bipartition on parallel line segments, ensuring diametral disks for cross-edges lie in the strip between segments, and surrounding each segment with witnesses to eliminate within-set edges.
• Obstructions: No complete 4-partite graph $K_{2,2,2,2}$ or $K_{3,3,3,3}$ admits a witness Gabriel realization. Any graph containing $K_{2,2,2,2}$ as an induced subgraph is not witness Gabriel.

Planarity is always retained, but the inclusion $\MST(P) \subseteq \GG^{-}(P,W) \subseteq \DT(P)$ (minimum spanning tree and Delaunay triangulation) does not in general hold unless $W\supseteq P$. Edges from the Euclidean minimum spanning tree may be removed by external witnesses.

4. Gabriel Quotient Rings: Construction and Algebraic Properties

Let $R$ be a prime right noetherian ring with right Krull dimension $|R|=n>1$, and let $Q$ be the Goldie quotient ring of $R$. For $0

• $$x_m = \{ p \in \operatorname{Spec} R : |R/p| = m \}$$ (the set of all $m$-full prime ideals).
• $C_m = \{ c \in R : |R/cR| < m \}$ (multiplicatively closed set of "small" regular elements).
• $g = \{ I \subseteq R : |R/I| < m \}$ (the $m$-Gabriel filter, a family of right ideals).

The $m$-Gabriel quotient ring $R(m)$ is the subring of $Q$ defined by

$$R(m) := \{ q \in Q : \exists\, I \in g \text{ such that } q I \subseteq R \}.$$

Properties:

• $R \subseteq R(m) \subseteq Q$ and $1 \in R(m)$ is the same as the identity of $R$ and $Q$.
• The group of units $U(R(m))$ satisfies $U(R(m)) \cap R = C_m$.
• If $x_m$ is a full set of $m$-prime ideals, $C_m$ is a right Ore set in $R$; $R(m)$ is the localization $R[C_m^{-1}]$ inside $Q$.

Alternatively, $R(m)/R$ consists precisely of the $g$-torsion elements of $Q/R$, i.e., $R(m)/R = t_g(Q/R) = \{ x \in Q/R : |xR| < m \}$ (Wangneo, 2023).

The construction generalizes classical Gabriel localization for commutative rings ($S$-localizations), and, in the noncommutative setting, yields genuinely intermediate rings between $R$ and $Q$ for Goldie dimension $>1$.

5. Illustrative Examples and Key Applications

Witness Gabriel Graphs

• For $W = \emptyset$, $\GG^{-}(P,W)$ is the complete graph; for $W = P$, it is the standard planar Gabriel graph with $n-1 \le |E| \le 3n - 8$.
• A single well-placed witness can remove a specified Gabriel edge.
• Points arranged in a hexagonal lattice may require $\frac{3}{4}n-o(n)$ witnesses to remove all edges, establishing a lower bound on editing capacity (Aronov et al., 2010).
• Witness Gabriel graphs have applications in graph drawing, computational geometry, and data mining by enabling models that reflect exclusion zones, interference, or negative-class data points.

Gabriel Quotient Rings

• For $R = k[x,y]$ (commutative polynomial ring, $|R|=2$) and $m=1$, $C_1=k[x,y]^\times=k^\times$, so $R(1)=R$; no non-trivial localization occurs.
• For $R = k[x,y,z]$ ($|R|=3$) and $m=2$, $C_2$ is the complement of all height $\ge2$ primes, yielding $R(2)=k[x,y,z][C_2^{-1}]$, strictly larger than $R$.
• In all noncommutative prime noetherian rings of Goldie dimension $>1$, the construction yields genuinely new intermediate rings.

Principal applications include the precise refinement of localizations in noncommutative ring theory and the control of edge sets in combinatorial geometry.

6. Open Problems and Future Directions

Several significant open problems and research directions remain:

• For witness Gabriel graphs, closing the gap between the $\frac{3}{4}n-o(n)$ and $n-1$ bounds for the number of witnesses required to remove all edges; designing subquadratic or output-sensitive construction algorithms in the worst case; determining computational complexity of minimal witness sets for realizing given graphs (e.g., NP-hardness status); and combinatorial characterization of witness Gabriel graphs beyond trees and bipartite graphs.
• For Gabriel quotient rings, understanding the structure of $R(m)$ for broader classes of rings, further connections between the Gabriel filter and Ore conditions, and the role of Krull-dimension bounds in controlling torsion submodules in noncommutative settings.

These directions underscore the flexibility and theoretical depth of Gabriel edited sets across discrete geometry and ring theory (Aronov et al., 2010, Wangneo, 2023).