Graph Characteristic Space: Embeddings & Invariants

• Graph characteristic space is a geometric and combinatorial construct that encodes intrinsic graph properties, including embeddings and algebraic invariants.
• It decomposes into cellules based on vertex partitions, providing a framework to study degenerations and stratified subvarieties in projective spaces.
• Key invariants like the minimum constraint dimension and matroid-based metrics reveal thresholds for rigidity and non-generic behavior in graph embeddings.

A graph characteristic space is a geometric, combinatorial, or algebraic structure associated with a graph that encodes intrinsic properties of graph embeddings, interactions, or invariants—often in a way that reveals deeper connections with algebraic geometry, rigidity theory, and matroid theory. In the context of the “picture spaces” introduced in the paper of graph varieties in high dimension, the graph characteristic space describes the collection of all point-and-line arrangements of a graph within an ambient projective space, together with its decomposition into algebraic subvarieties and the associated combinatorial and algebraic invariants that distinguish its irreducible components and dimension-dependent behavior (Enkosky et al., 2010).

1. Definition and Fundamental Objects

The central geometric structure is the picture space $X^d(G)$ of a finite graph $G=(V,E)$ in a projective space $\mathbb{P}^d$. A picture of $G$ consists of an assignment:

• to each vertex $v \in V$, a point $P(v)\in\mathbb{P}^d$;
• to each edge $e \in E$, a line $P(e)\subset \mathbb{P}^d$, subject to the incidence condition: if $v$ is incident to $e$, then $P(v)\in P(e)$.

$X^d(G)$ realizes the set of all such assignments as an algebraic subset of a product of Grassmannians defined by the containment (incidence) constraints.

Two further graph characteristic spaces appear as restrictions or projections:

• the picture variety $V(G)$, the Zariski closure of the locus of generic pictures (in which the points $P(v)$ are pairwise distinct);
• the slope variety, the image of $V(G)$ under the projection recording the directions of the edge-lines.

These spaces are algebraic varieties encoding core combinatorial and geometric properties of $G$ as embedded in the ambient space.

2. Decomposition into Cellules

$X^d(G)$ admits a natural decomposition into smooth, locally closed, quasiprojective subvarieties called cellules, indexed by set partitions $\pi$ of $V$:

$${\displaystyle X_\pi(G) = \left\{ P\in X^d(G) \mid P(v)=P(w)\ \Leftrightarrow\ v\sim_\pi w \right\} }$$

Here, vertices in the same block of $\pi$ are required to collapse to the same point; vertices in different blocks are mapped to distinct points.

Each cellule $X_\pi(G)$ is a fiber bundle over $(\mathbb{P}^d)^{|\pi|}$ (the positions of the blocks), with fiber $(\mathbb{P}^{d-1})^{\delta(\pi, G)}$, where $\delta(\pi, G)$ counts the number of edges whose endpoints are within the same block. Its dimension is:

$${\displaystyle \dim X_\pi(G) = d\cdot|\pi| +(d-1)\cdot \delta(\pi,G) }$$

Particular cases include:

• The discrete cellule ($\pi$ the partition into singletons): all points remain distinct—this corresponds to the generic embedding locus.
• The indiscrete cellule (a single block): all vertices coincide.

Cellules not only facilitate the stratification of $X^d(G)$ but also index key geometric degenerations (vertex collapses) in families of graph embeddings.

3. Irreducible Components and the Cellule Partial Order

The closures of certain cellules provide the irreducible components of $X^d(G)$. The cellule closure inclusion relation defines the cellule order $\prec_{G,d}$ on the partition lattice:

$${\displaystyle \pi \prec_{G,d} \sigma \iff X_\pi(G) \subseteq \overline{X_\sigma(G)} }$$

Only the maximal cellules (with respect to this order) yield the irreducible components.

For the complete graph $K_n$ and $d\ge 3$, the irreducible components are precisely the cellules corresponding to partitions without blocks of size two; i.e., blocks must either be singletons or have size at least three. This combinatorial criterion for maximality has far-reaching implications for the structure of the characteristic space in general graphs, reducing component analysis to a question of set partition refinement and incidence geometry.

4. Combinatorial Formulas and the Minimum Constraint Dimension

A central invariant of the characteristic space is the minimum constraint dimension $\operatorname{mcd}(G)$, the smallest $d$ such that $X^d(G)$ fails to be irreducible—i.e., the ambient dimension at which collapsed cellules become "visible" as separate irreducible components, and nontrivial directional constraints on edge lines emerge.

$\operatorname{mcd}(G)$ is given by:

$${\displaystyle \operatorname{mcd}(G) = \min \left\{ d\in\mathbb{N} \mid \exists\,A\subset E,\, d \cdot \mathrm{nul}(A) \ge |A| \right\} = \min_{A\ne\emptyset} \left\lceil \frac{|A|}{\mathrm{nul}(A)} \right\rceil }$$

where $\mathrm{nul}(A) = |A| - \mathrm{rank}(A)$ denotes nullity with respect to the graphic matroid of $G$. The existence of flats (maximal subgraphs without adding rank) and ear decompositions (decompositions into cycles) refine this formula, connecting $\operatorname{mcd}(G)$ to deep combinatorial invariants (Tutte polynomial, 2-connected induced subgraphs).

For cycles $C_n$, $\operatorname{mcd}(C_n) = n$.

5. Applications and Interdisciplinary Impact

Characteristic space results have significant implications in various fields:

• Rigidity theory: Stratification by cellules and calculation of $\operatorname{mcd}(G)$ delineate the dimension thresholds where edge direction constraints (and hence rigidity-type properties) first arise. For $d=2$, the emergence of nontrivial components aligns with the minimal rigidity circuits of Laman.
• Combinatorial geometry: The decomposition into cellules and component analysis translates enumerative problems and degenerations of graph embeddings into poset-theoretic and matroidal language.
• Algebraic geometry: The passage from planar cases to arbitrary dimensions enables the application of modern tools (Grassmannian embeddings, Poincaré and Tutte polynomials) and situates graph varieties within the broader theory of moduli spaces.
• Degeneration phenomena: Understanding which vertex collapse patterns (encoded by non-generic cellules) persist in high dimensions informs the paper of limits and specializations in families of algebraic varieties associated to graphs.

A major implication is the large gap that can occur between classical invariants like girth and the true minimum constraint dimension, highlighting the subtlety of constraints induced by graph structure in high-dimensional settings.

6. Connections to Poset and Matroid Theory

The cellule order structure constitutes an instance of a poset arising from geometric stratification, whose maximal elements index algebraic components. The explicit use of graphic matroids in component and constraint calculations underscores the matroidal underpinnings of the entire theory, tying the geometry of the characteristic space tightly to independence and basis structure in $G$.

Emerging problems concern the full classification of cellule orders for various $G$ and $d$, and the enumeration of posets arising from characteristic spaces—a line of inquiry with implications for both combinatorics and algebraic geometry.

7. Summary Table of Key Formulas and Invariants

Concept	Formula/Description	Context
Cellule dimension	$\dim X_\pi(G) = d\,|\pi| + (d-1)\,\delta(\pi,G)$	Structure of $X^d(G)$ stratification
Minimum constraint dimension	$$\displaystyle\operatorname{mcd}(G) = \min_{A\neq\varnothing} \left\lceil \frac{|A|}{\mathrm{nul}(A)} \right\rceil$$	Onset of non-generic components
Partial order on partitions	$$\pi \prec_{G,d} \sigma \iff X_\pi(G) \subseteq \overline{X_\sigma(G)}$$	Irreducible component identification

The paper of graph characteristic spaces developed in the context of picture spaces creates a systematic and unifying language to encode degenerations, constraints, and rigidity phenomena for graph embeddings in projective geometry. The use of algebraic stratification, matroid invariants, and combinatorial data bridges discrete, geometric, and algebraic perspectives, generating tools for the analysis of both fine and global properties of families of graph varieties (Enkosky et al., 2010).