Graph Theory

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

FLOW ALGEBRA
TITLE: Graph Theory — Foundational Ontology
DOMAIN: Discrete Mathematics / Graph Theory
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/graph-theory.html
VERSION: 1.0
DATE: 2026-10-11
AUTHORS: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org
STATUS: Foundational mathematical model; source page not independently verified.

1. PURPOSE
Represent graphs, their properties, transformations and algorithms as typed, verifiable semantic flows for mathematical reasoning, graph databases and AI knowledge systems.

2. TYPED ENTITIES
Graph := (V:Set[Vertex], E:Set[Edge], directed:Boolean, loops_allowed:Boolean, parallel_edges_allowed:Boolean)
Vertex := identifier with optional attributes
UndirectedEdge := unordered pair/multiset of endpoints from V
DirectedEdge := ordered pair (source,target) in V x V
SimpleGraph := undirected graph with no loops or parallel edges
Digraph := graph with directed edges
WeightedGraph := graph with weight:E -> Number (or specified weight domain)
Subgraph := graph H with V(H) subseteq V(G) and E(H) subseteq E(G), endpoints retained
Path := vertex-edge sequence with incident consecutive edges; a simple path has no repeated vertices
Walk := alternating vertex-edge sequence respecting incidence and direction
Cycle := closed path with distinct internal vertices (conventions for loops and length two must be specified)
Tree := connected acyclic undirected graph
Forest := acyclic undirected graph
Component := maximal connected subgraph (weak/strong variants for digraphs)
Adjacency := incidence relation between vertices under an edge
Degree := number of incident edge ends; loops contribute two in undirected graphs

3. CORE FLOW OPERATIONS
CREATE_GRAPH(V,E,kind) -> Graph after endpoint and multiplicity validation
ADD_VERTEX(G,v) -> G' with v included
ADD_EDGE(G,e) -> G' with e included when endpoints and graph constraints permit
REMOVE_VERTEX(G,v) -> induced deletion of v and all incident edges
REMOVE_EDGE(G,e) -> G' without specified edge instance
NEIGHBOURS(G,v) -> Set[Vertex] (in/out variants for digraphs)
DEGREE(G,v) -> NonNegativeInteger
REACHABLE(G,u,v) -> Boolean
CONNECTED_COMPONENTS(G) -> Partition[V] for undirected graphs
SHORTEST_PATH(G,u,v,w) -> path or NONE; algorithm preconditions depend on weights
TRAVERSE_BFS(G,s) -> ordered visit sequence and optional search tree
TRAVERSE_DFS(G,s) -> ordered visit sequence and optional search forest
TOPOLOGICAL_SORT(G) -> ordering or CYCLE_FOUND for finite digraphs
SPANNING_TREE(G) -> tree spanning V when undirected G is connected

4. FORMAL AXIOMS AND IDENTITIES
A1: For every edge e, all endpoints of e belong to V(G).
A2: For a finite undirected multigraph, sum(v in V) degree(v) = 2|E|, counting loops twice.
A3: For a finite digraph, sum(v) indegree(v) = sum(v) outdegree(v) = |E|.
A4: For an undirected graph, connected(u,v) is reflexive, symmetric and transitive.
A5: A finite undirected graph is a tree iff it is connected and acyclic.
A6: Every finite tree with n vertices has n-1 edges (including the one-vertex tree).
A7: For finite simple undirected graphs, 0 <= |E| <= n(n-1)/2.
A8: An undirected graph is bipartite iff it contains no odd cycle.
A9: A finite directed graph has a topological ordering iff it is acyclic.
A10: Isomorphic graphs preserve adjacency, degree multiset, connectivity and cycle structure.

5. RELATIONSHIPS
Graph HAS_VERTEX Vertex
Graph HAS_EDGE Edge
Edge CONNECTS Vertex
DirectedEdge HAS_SOURCE Vertex
DirectedEdge HAS_TARGET Vertex
Path TRAVERSES Edge
Cycle IS_A ClosedWalk
Tree IS_A ConnectedAcyclicUndirectedGraph
Forest IS_A AcyclicUndirectedGraph
Component IS_MAXIMAL_CONNECTED_SUBGRAPH_OF Graph
Subgraph IS_CONTAINED_IN Graph
Graph IS_ISOMORPHIC_TO Graph when a bijective adjacency-preserving mapping exists

6. INFERENCE RULES
R1: TREE(G) and FINITE(G) => |E(G)| = |V(G)| - 1.
R2: UNDIRECTED(G) and CONNECTED(G) and ACYCLIC(G) => TREE(G).
R3: DIRECTED(G) and FINITE(G) and ACYCLIC(G) => EXISTS TOPOLOGICAL_ORDER(G).
R4: HAS_ODD_CYCLE(G) => NOT BIPARTITE(G) for undirected G.
R5: ISOMORPHIC(G,H) and CONNECTED(G) => CONNECTED(H).
R6: REACHABLE(G,u,v) and REACHABLE(G,v,w) => REACHABLE(G,u,w), for consistent directed/undirected reachability.
R7: UNDIRECTED(G) and DEGREE_SUM(G)=2|E(G)| => HANDSHAKING_INVARIANT_SATISFIED(G); not a complete validity proof.

7. ALGORITHMIC SEMANTICS
BFS: unweighted shortest-path distances by edge count from a source; O(|V|+|E|) with adjacency lists.
DFS: reachability, cycle analysis and traversal; O(|V|+|E|) with adjacency lists.
Dijkstra: shortest paths for non-negative edge weights; implementation-dependent complexity.
Bellman-Ford: handles negative weights; reports reachable negative cycles.
Kruskal/Prim: minimum spanning trees for weighted undirected connected graphs.
Topological sort: finite directed acyclic graphs only.

8. VALIDATION
V1: All edge endpoints exist in V.
V2: Directedness and edge multiplicity conform to graph kind.
V3: Adjacency representation is consistent with the edge collection.
V4: Degree sums match the appropriate finite-graph identity.
V5: Tree claims require connectivity and acyclicity.
V6: Shortest-path outputs respect path incidence and stated weight assumptions.
V7: Algorithm outputs include provenance, graph version and preconditions.
V8: Do not infer graph isomorphism solely from matching degree sequences.

9. SEMANTIC_EQUIVALENCE
Graph isomorphism: G ~= H iff a vertex bijection preserves and reflects edge incidence (including directions/multiplicities when applicable).
Representation equivalence: adjacency lists, adjacency matrices and edge lists denote the same graph only after node identifiers, directedness, loops and multiplicities are normalised.
Terminology: 'node' ~= 'vertex' in graph-theoretic context; 'link' ~= 'edge' only when incidence semantics match.
Semantic similarity is not mathematical equivalence.

10. CONFIDENCE
Mathematical axioms: 1.00 within declared graph conventions and preconditions.
Validated algorithmic results: 1.00 conditional on correct implementation and inputs.
Extracted real-world graph assertions: confidence must derive from evidence, provenance and freshness; never assign a universal numeric score.
Unverified claims: UNKNOWN rather than assumed true.

11. FLOW_HASH
Canonicalise UTF-8 content, ordered sections, typed entity definitions, operation signatures, preconditions and version metadata.
FLOW_HASH := SHA256(canonical_serialisation); computed by deployment pipeline, not fabricated in this document.

12. UPDATE_MODEL
On vertex/edge mutation: increment graph version, invalidate affected reachability/path/component caches, rerun applicable invariants and recalculate FLOW_HASH.
On ontology revision: retain previous version and record change provenance.
On source revision: review definitions and regenerate the model only after checking source changes.

13. AI_CONSUMPTION
INPUT: typed graph specification, evidence metadata and query.
PIPELINE: parse -> validate -> select graph semantics -> execute sound inference/algorithm -> verify output -> return traceable result.
OUTPUT: result, assumptions, supporting vertices/edges or proof trace, graph version, confidence and error/unknown status.
GUARDRAILS: do not conflate directed and undirected reachability; distinguish paths from walks; reject unsupported shortest-path assumptions; report missing or contradictory data.

END FLOW ALGEBRA