home / skills / plurigrid / asi / graph-grafting

graph-grafting skill

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This skill applies graph grafting concepts to attach rooted trees to graphs, enabling advanced topology operations and combinatorial reasoning.

npx playbooks add skill plurigrid/asi --skill graph-grafting

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---
name: graph-grafting
description: Graph Grafting Skill
version: 1.0.0
---

# Graph Grafting Skill

**Trit**: 0 (ERGODIC - Coordinator)
**GF(3) Triad**: `queryable (-1) ⊗ graftable (0) ⊗ derangeable (+1) = 0`

## Overview

Combinatorial complex operations replacing GraphQL with pure graph theory:

| Operation | Trit | Description |
|-----------|------|-------------|
| **Queryable** | -1 | Tree-shape decision via bag decomposition |
| **Colorable** | 0 | GF(3) 3-coloring via sheaf |
| **Derangeable** | +1 | Permutations with no fixed points |
| **Graftable** | 0 | Attach rooted tree at vertex |

## Mathematical Foundation

**Grafting** = attaching a rooted tree T at vertex v of graph G:

```
Graft(T, v, G) → G' where:
  - V(G') = V(G) ∪ V(T)
  - E(G') = E(G) ∪ E(T) ∪ {(v, root(T))}
  - Adhesion = shared labels at attachment point
```

## Quadrant Chart: Colorable × Derangeable

```
        Balanced (GF3=0)
              │
    Q2        │        Q1 ← OPTIMAL
  Identity    │    PR#18, Knight Tour
              │    SICM Galois
──────────────┼──────────────
    Q3        │        Q4
  Deadlock    │    Phase Trans
              │
        Fixed Points → Derangement
```

## Usage

```julia
using .GraphGrafting

c = GraftComplex(UInt64(1069))

# Build PR tree
root = GraftNode(:pr18, Int8(0), :golden, 0)
alice = GraftNode(:alice, Int8(-1), :baseline, 1)
bob = GraftNode(:bob, Int8(1), :original, 1)

# Graft nodes
graft!(c, root, :none, String[])
graft!(c, alice, :pr18, ["aptos-wallet-mcp"])
graft!(c, bob, :pr18, ["aptos-wallet-mcp"])

# Operations
tree_shape(c)           # Queryable
trit_partition(c)       # Colorable  
derange!(c)             # Derangeable
compose(c1, c2, :vertex) # Graftable

# Verify
verify_gf3(c)  # → (conserved=true, sum=0)
```

## Neighbors

### High Affinity
- `three-match` (-1): Graph coloring verification
- `derangeable` (+1): No fixed points
- `bisimulation-game` (-1): Attacker/Defender

### Example Triad
```yaml
skills: [graph-grafting, three-match, derangeable]
sum: (0) + (-1) + (+1) = 0 ✓ CONSERVED
```

## References

- Joyal, Combinatorial Species (1981)
- Flajolet & Sedgewick, Analytic Combinatorics (2009)
- Topos Institute, Observational Bridge Types



## Scientific Skill Interleaving

This skill connects to the K-Dense-AI/claude-scientific-skills ecosystem:

### Graph Theory
- **networkx** [○] via bicomodule
  - Graph manipulation and algorithms

### Bibliography References

- `graph-theory`: 38 citations in bib.duckdb



## SDF Interleaving

This skill connects to **Software Design for Flexibility** (Hanson & Sussman, 2021):

### Primary Chapter: 1. Flexibility through Abstraction

**Concepts**: combinators, compose, parallel-combine, spread-combine, arity

### GF(3) Balanced Triad

```
graph-grafting (−) + SDF.Ch1 (+) + [balancer] (○) = 0
```

**Skill Trit**: -1 (MINUS - verification)

### Secondary Chapters

- Ch4: Pattern Matching
- Ch10: Adventure Game Example

### Connection Pattern

Combinators compose operations. This skill provides composable abstractions.
## Cat# Integration

This skill maps to **Cat# = Comod(P)** as a bicomodule in the equipment structure:

```
Trit: 0 (ERGODIC)
Home: Prof
Poly Op: ⊗
Kan Role: Adj
Color: #26D826
```

### GF(3) Naturality

The skill participates in triads satisfying:
```
(-1) + (0) + (+1) ≡ 0 (mod 3)
```

This ensures compositional coherence in the Cat# equipment structure.

Overview

This skill implements graph grafting primitives for composable combinatorial operations on graphs. It replaces GraphQL-style queries with pure graph-theoretic transformations: attaching rooted trees, GF(3) color checks, tree-shape decomposition, and derangement operations. The focus is on composability, verification, and preserving algebraic invariants across graft operations.

How this skill works

The core operation Graft(T, v, G) attaches a rooted tree T at vertex v of graph G, merging vertex and edge sets and linking via the root. The skill exposes utilities to compute tree-shape decompositions (queryable), run GF(3) 3-coloring checks (colorable), and perform derangement permutations (derangeable). Higher-level combinators let you compose complexes, verify GF(3) conservation, and partition by trit values for coordinated workflows.

When to use it

Attach or merge tree-structured data into an existing graph while preserving labels and adhesion semantics.
Verify combinatorial invariants such as GF(3) sums after graph composition.
Model decision trees or bag decompositions where tree-shaped queries replace schema queries.
Apply derangement-based permutations to test fixed-point-free transformations.
Compose multiple graph fragments safely using graftable combinators.

Best practices

Keep attachment points (vertices) clearly labeled so adhesion semantics are deterministic.
Use trit partitions (queryable/colorable/derangeable) to separate verification, coloring, and permutation concerns.
Run verify_gf3 after composition to ensure conserved algebraic invariants before downstream processing.
Prefer small rooted trees for incremental grafting when experimenting to simplify debugging.
Compose via provided combinators rather than manipulating low-level sets to retain categorical coherence.

Example use cases

Building a PR tree by grafting contributor nodes and verifying GF(3) conservation across the complex.
Replacing GraphQL queries with tree-shape decision logic using bag decompositions for read-heavy analytics.
Constructing test graphs that exercise derangement properties (no fixed points) to validate permutation-based algorithms.
Composing heterogenous graph fragments from different pipelines and verifying adhesive label consistency at graft points.
Using trit_partition to route graph elements into color-checking, querying, or derangement pipelines for staged processing.

FAQ

What does GF(3) verification check?

It verifies that the ternary color sum (mod 3) across the complex is conserved and reports whether the sum equals the expected balanced value.

How do I attach a tree without altering existing labels?

Label the root and target vertex explicitly and use the graft combinator which merges sets and links via the root while preserving existing labels and adhesion semantics.