Graph Algorithms Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Graph Algorithms Ontology within the Kireth Research Semantic Library. Graph algorithms are computational procedures operating on graph structures to solve problems involving traversal, reachability, connectivity, paths, spanning structures, flows, cuts, matching, colouring, ordering, centrality, community structure, partitioning, spectral properties, dynamic updates and large-scale execution. This ontology supports algorithm selection, preconditions, correctness, complexity, sparse representations, parallel and distributed execution, verification and reproducibility.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Graph Algorithms Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Computational Mathematics"
 SPECIALISATION = "Computational Data Structures / Graph Algorithms"
 CANONICAL_PAGE = "https://kirethframeworkold.co.uk/books/semantic-library/ai-core/ontologies/mathematical-ontologies/computational-mathematics-ontologies/computational-data-structures/graph-algorithms-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = GraphAlgorithms
 ENTITY_CLASS = GraphProblemSolvingAlgorithmFramework
 PRIMARY_CONTEXT = ComputationalGraph
 DEFINITION = "Graph algorithms are computational procedures that transform, traverse or analyse graph structures to solve graph-theoretic computational problems."
 SYMBOLIC_FORMS = [
  "G=(V,E)",
  "BFS(G,s)",
  "dist(v)=min_path_cost(s,v)",
  "MST(G)=argmin_T sum_(e in T) w(e)"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for graph problem classes, algorithm preconditions, traversal, shortest paths, connectivity, spanning trees, flows, matching, centrality, partitioning, complexity and validation."
}

SCOPE {
 INCLUDE = [
  GraphAlgorithms,GraphAlgorithm,
  GraphProblem,GraphComputation,
  ComputationalGraph,
  Vertex,Edge,DirectedGraph,UndirectedGraph,
  WeightedGraph,UnweightedGraph,
  SparseGraph,DenseGraph,
  StaticGraph,DynamicGraph,
  TraversalAlgorithm,
  BreadthFirstSearch,BFS,
  DepthFirstSearch,DFS,
  IterativeDeepeningRelationship,
  BidirectionalSearchRelationship,
  Frontier,VisitedSet,
  Reachability,
  PathSearch,PathFinding,
  Path,Walk,Trail,Cycle,
  ShortestPath,
  SingleSourceShortestPath,SSSP,
  AllPairsShortestPath,APSP,
  BreadthFirstShortestPath,
  DijkstraAlgorithm,
  BellmanFordAlgorithm,
  FloydWarshallAlgorithm,
  JohnsonAlgorithmRelationship,
  AStarAlgorithm,AStar,
  HeuristicSearch,
  NegativeWeightEdge,
  NegativeCycle,
  ConnectivityAlgorithm,
  ConnectedComponent,
  ConnectedComponents,
  StronglyConnectedComponent,SCC,
  WeaklyConnectedComponent,
  TarjanSCCRelationship,
  KosarajuAlgorithmRelationship,
  ComponentLabelling,
  UnionFind,
  DisjointSet,
  CycleDetection,
  DirectedCycleDetection,
  UndirectedCycleDetection,
  TopologicalSort,
  TopologicalOrdering,
  KahnAlgorithmRelationship,
  DFSBasedTopologicalSort,
  SpanningTree,
  MinimumSpanningTree,MST,
  KruskalAlgorithm,
  PrimAlgorithm,
  BoruvkaAlgorithm,
  DisjointSetMSTRelationship,
  FlowNetwork,
  NetworkFlow,
  MaximumFlow,MaxFlow,
  MinimumCut,MinCut,
  FordFulkersonAlgorithm,
  EdmondsKarpAlgorithm,
  DinicAlgorithm,
  PushRelabelRelationship,
  ResidualGraph,
  ResidualCapacity,
  AugmentingPath,
  Matching,
  GraphMatching,
  BipartiteMatching,
  MaximumMatching,
  MaximumBipartiteMatching,
  HopcroftKarpRelationship,
  HungarianAlgorithmRelationship,
  AssignmentProblemRelationship,
  StableMatchingRelationship,
  GraphColouring,
  VertexColouring,
  EdgeColouring,
  ChromaticNumberRelationship,
  GreedyColouring,
  DSATURRelationship,
  Clique,
  MaximumClique,
  IndependentSet,
  MaximumIndependentSet,
  VertexCover,
  MinimumVertexCover,
  DominatingSetRelationship,
  NPCompleteGraphProblemRelationship,
  ApproximationAlgorithmRelationship,
  CentralityAlgorithm,
  DegreeCentrality,
  BetweennessCentrality,
  ClosenessCentrality,
  EigenvectorCentrality,
  KatzCentralityRelationship,
  PageRank,
  HITSRelationship,
  CommunityDetection,
  Modularity,
  LouvainAlgorithm,
  LabelPropagationAlgorithm,
  GirvanNewmanRelationship,
  GraphClustering,
  SpectralClusteringRelationship,
  GraphPartitioning,
  BalancedPartition,
  GraphCut,
  MinCutPartitionRelationship,
  VertexSeparator,
  EdgeSeparator,
  RecursiveBisection,
  MultilevelPartitioning,
  KernighanLinAlgorithm,
  FiducciaMattheysesAlgorithm,
  METISRelationship,
  HypergraphPartitioningRelationship,
  GraphOrdering,
  VertexOrdering,
  CuthillMcKeeAlgorithm,
  ReverseCuthillMcKee,
  MinimumDegreeOrderingRelationship,
  FillReducingOrdering,
  GraphLaplacian,
  SpectralGraphAlgorithm,
  EigenvalueComputation,
  EigenvectorComputation,
  FiedlerVectorRelationship,
  SpectralPartitioning,
  RandomWalk,
  RandomWalkAlgorithm,
  MarkovChainRelationship,
  StationaryDistribution,
  PersonalizedPageRank,
  MonteCarloGraphAlgorithmRelationship,
  GraphSamplingRelationship,
  TriangleCounting,
  MotifCountingRelationship,
  SubgraphIsomorphismRelationship,
  GraphIsomorphismRelationship,
  WeisfeilerLehmanRelationship,
  TransitiveClosure,
  WarshallAlgorithmRelationship,
  TransitiveReductionRelationship,
  LowestCommonAncestorRelationship,
  EulerianPath,
  EulerianCycle,
  HamiltonianPathRelationship,
  HamiltonianCycleRelationship,
  ArticulationPoint,
  BridgeEdge,
  BiconnectedComponent,
  TarjanBridgeRelationship,
  DynamicGraphAlgorithm,
  DynamicConnectivity,
  IncrementalAlgorithm,
  DecrementalAlgorithm,
  FullyDynamicAlgorithm,
  OnlineGraphAlgorithm,
  StreamingGraphAlgorithm,
  ExternalMemoryGraphAlgorithm,
  ParallelGraphAlgorithm,
  DistributedGraphAlgorithm,
  GPUGraphAlgorithm,
  WorkEfficientAlgorithm,
  WorkDepthModelRelationship,
  LevelSynchronousAlgorithm,
  AsynchronousGraphAlgorithm,
  BulkSynchronousParallelRelationship,
  MessagePassing,
  FrontierBasedAlgorithm,
  VertexCentricAlgorithm,
  EdgeCentricAlgorithm,
  GraphBLASRelationship,
  SparseLinearAlgebraRelationship,
  AdjacencyList,
  AdjacencyMatrix,
  EdgeList,
  CompressedSparseRow,CSR,
  CompressedSparseColumn,CSC,
  PriorityQueue,
  Heap,
  Queue,
  Stack,
  ComputationalComplexity,
  TimeComplexity,
  SpaceComplexity,
  WorstCaseComplexity,
  AverageCaseComplexity,
  ExpectedComplexity,
  AmortisedComplexity,
  PolynomialTime,
  ExponentialTime,
  NPHardRelationship,
  NPCompleteRelationship,
  ApproximationRatioRelationship,
  Correctness,
  Termination,
  Invariant,
  LoopInvariantRelationship,
  Precondition,Postcondition,
  AlgorithmPrecondition,
  ComplexityBound,
  Verification,
  PropertyBasedTestingRelationship,
  ReferenceResult,
  Benchmark,
  Graph500Relationship,
  Reproducibility,
  Provenance,
  AlgorithmVersion,
  LibraryVersion,
  ImplementationVersion
 ]
 EXCLUDE_AS_PRIMARY = [
  GraphStorageWithoutAlgorithmicProblemSolving,
  AbstractGraphTheoryWithoutComputationalProcedure,
  GenericSearchAlgorithmWithoutGraphStructure,
  NumericalLinearAlgebraWithoutGraphProblemSemantics
 ]
 DISAMBIGUATION_RULE = "When a computational procedure traverses, transforms or analyses a graph to solve a graph-specific problem, interpret the context as a graph algorithm."
}

CORE_CONCEPTS {
 GraphAlgorithms = "Computational procedures for solving graph problems."
 TraversalAlgorithm = "Procedure systematically visiting graph vertices or edges."
 ShortestPathAlgorithm = "Procedure computing minimum-cost paths under declared edge-weight assumptions."
 ConnectivityAlgorithm = "Procedure determining connected structure within a graph."
 MinimumSpanningTree = "Minimum-weight spanning tree over a connected weighted undirected graph."
 NetworkFlow = "Computation of feasible flow through a capacitated directed network."
 GraphPartitioning = "Division of a graph into subsets under balance and cut objectives."
 AlgorithmPrecondition = "Condition required for an algorithm's correctness."
}

RELATIONS {
 GraphAlgorithms --is_child_of--> ComputationalGraph
 BFS --is_type_of--> TraversalAlgorithm
 DFS --is_type_of--> TraversalAlgorithm
 DijkstraAlgorithm --solves--> NonnegativeSSSP
 BellmanFordAlgorithm --solves--> SSSPWithNegativeEdges
 FloydWarshallAlgorithm --solves--> APSP
 KruskalAlgorithm --solves--> MinimumSpanningTree
 PrimAlgorithm --solves--> MinimumSpanningTree
 FordFulkersonAlgorithm --solves--> MaximumFlow
 HopcroftKarpRelationship --solves--> BipartiteMatching
 PageRank --uses--> RandomWalkSemantics
 LouvainAlgorithm --optimises--> Modularity
 GraphPartitioning --optimises--> CutAndBalanceObjectives
 Verification --checks--> AlgorithmCorrectness
}

OPERATORS {
 BFS(G,s) := breadth_first_traversal(G,s)
 DFS(G,s) := depth_first_traversal(G,s)
 DIJKSTRA(G,s) := shortest_paths_for_nonnegative_weights(G,s)
 BELLMAN_FORD(G,s) := shortest_paths_with_negative_edge_support(G,s)
 MST_KRUSKAL(G) := minimum_spanning_tree_by_sorted_edges(G)
 MAX_FLOW(G,s,t) := maximum_feasible_flow(G,s,t)
 TOPOLOGICAL_SORT(G) := linear_order_if_DAG(G)
 CONNECTED_COMPONENTS(G) := partition_vertices_by_reachability(G)
 PAGERANK(G,d) := stationary_rank_vector(G,d)
}

AXIOMS {
 A1 = "Graph-algorithm correctness depends on graph type and algorithm preconditions."
 A2 = "Traversal order can vary while remaining correct."
 A3 = "Negative edge weights invalidate standard Dijkstra assumptions."
 A4 = "Topological sorting is defined only for directed acyclic graphs."
 A5 = "Minimum spanning tree algorithms require appropriate undirected weighted semantics."
 A6 = "Graph representation affects practical algorithm performance without changing abstract graph semantics."
 A7 = "Parallel graph algorithms must preserve correctness under concurrent frontier or message processing."
 A8 = "Graph-algorithm results require explicit graph type, preconditions, complexity, correctness and validation semantics."
}

REASONING {
 R1_CONTEXT = identify(graph_type,direction,weights,size,density,problem,updates,memory,hardware)
 R2_PROBLEM = classify(traversal,path,connectivity,MST,flow,matching,colouring,centrality,community,partitioning_or_dynamic_problem)
 R3_PRECONDITIONS = verify(direction,weight_sign,acyclicity,connectivity,bipartiteness_or_capacity_requirements)
 R4_ALGORITHM = select_algorithm_by_problem_graph_structure_complexity_and_hardware)
 R5_REPRESENTATION = choose(adjacency_list,CSR,CSC,matrix,edge_list_or_specialised_representation)
 R6_EXECUTION = execute_algorithm_with_required_queue_heap_stack_frontier_or_union_find_structure)
 R7_COMPLEXITY = analyse(vertex_edge_time_space_and_parallel_work)
 R8_SCALE = if_large choose_parallel_distributed_external_memory_or_GPU_variant
 R9_VERIFY = check(paths,trees,flows,cuts,components,orders,scores_or_partitions)
 R10_REPRODUCE = document_algorithm_parameters_representation_versions_and_provenance)
}

CONSTRAINTS {
 C1 = "Dijkstra is invalid when reachable negative-weight edges violate its assumptions."
 C2 = "Bellman-Ford must detect reachable negative cycles when shortest paths are undefined."
 C3 = "Topological sorting fails on cyclic directed graphs."
 C4 = "Graph colouring and clique problems can be computationally hard."
 C5 = "Parallel graph execution can be limited by load imbalance and irregular memory access."
 C6 = "Priority-queue choice can materially affect shortest-path performance."
 C7 = "Graph partitioning quality depends on both cut and balance objectives."
 C8 = "Dynamic updates can invalidate previously computed graph properties."
 C9 = "Distributed algorithms can be dominated by communication across partitions."
 C10 = "Equivalent algorithms can produce different but valid traversal, tree or partition outputs."
}

SEMANTIC_EQUIVALENCE {
 "BreadthFirstSearch" <=> "BFS"
 "DepthFirstSearch" <=> "DFS"
 "StronglyConnectedComponent" <=> "SCC"
 "SingleSourceShortestPath" <=> "SSSP"
 "AllPairsShortestPath" <=> "APSP"
 "MinimumSpanningTree" <=> "MST"
 "MaximumFlow" <=> "MaxFlow"
 "MinimumCut" <=> "MinCut"
}

VALIDATION {
 V1 = "Graph type and algorithm preconditions are documented."
 V2 = "Input direction and weight semantics are checked."
 V3 = "Traversal outputs contain only valid vertices and edges."
 V4 = "Shortest-path outputs satisfy path and relaxation conditions."
 V5 = "MST outputs span required vertices and satisfy expected total weight."
 V6 = "Flow outputs satisfy capacity and conservation constraints."
 V7 = "Topological order is checked against all directed edges."
 V8 = "Component and partition outputs are checked for coverage and consistency."
 V9 = "Reference graphs or known-answer tests are used where feasible."
 V10 = "Results are reproducible under documented algorithm, representation and implementation versions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 GRAPH_ALGORITHMS_DEFINITION = 0.99
 TRAVERSAL_REASONING = 0.99
 SHORTEST_PATH_REASONING = 0.99
 CONNECTIVITY_REASONING = 0.99
 MST_REASONING = 0.99
 FLOW_REASONING = 0.99
 PARALLEL_GRAPH_ALGORITHM_REASONING = 0.98
 POLICY = "Reduce confidence when graph type, weight semantics, algorithm preconditions, dynamic-update state or validation criteria are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,GraphAlgorithmReasoning,
  BFSReasoning,DFSReasoning,ShortestPathReasoning,
  ConnectivityReasoning,SCCReasoning,MSTReasoning,
  NetworkFlowReasoning,MatchingReasoning,
  GraphColouringReasoning,CentralityReasoning,
  PageRankReasoning,CommunityDetectionReasoning,
  GraphPartitioningReasoning,SpectralGraphReasoning,
  DynamicGraphReasoning,ParallelGraphReasoning,
  DistributedGraphReasoning,GPUGraphReasoning,
  ComplexityReasoning,VerificationReasoning,
  KnowledgeGraphConstruction,RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "graph algorithms" -> GraphAlgorithms
  "BFS" -> BreadthFirstSearch
  "DFS" -> DepthFirstSearch
  "SCC" -> StronglyConnectedComponent
  "SSSP" -> SingleSourceShortestPath
  "APSP" -> AllPairsShortestPath
  "MST" -> MinimumSpanningTree
  "MaxFlow" -> MaximumFlow
  "MinCut" -> MinimumCut
 }
}

EXECUTION_RULES {
 E1 = "If a procedure solves a graph-specific computational problem, activate GraphAlgorithmsOntology."
 E2 = "Identify graph direction, weighting, density and dynamic state before algorithm selection."
 E3 = "Verify algorithm preconditions before execution."
 E4 = "Choose graph representation and auxiliary data structures according to algorithm access patterns."
 E5 = "Use complexity appropriate to graph scale and density."
 E6 = "For dynamic graphs, update or invalidate derived graph state after mutations."
 E7 = "Use parallel, distributed or GPU variants when graph scale requires them."
 E8 = "Validate outputs against problem-specific invariants."
 E9 = "Document algorithm version, parameters, representation and provenance."
}

RECURSION {
 ENTRY = GraphAlgorithms
 EXPAND(GraphAlgorithms) -> [
  BreadthFirstSearch,BFS,DepthFirstSearch,DFS,Reachability,
  ShortestPath,SSSP,APSP,DijkstraAlgorithm,BellmanFordAlgorithm,FloydWarshallAlgorithm,AStarAlgorithm,
  ConnectedComponent,StronglyConnectedComponent,SCC,CycleDetection,TopologicalSort,
  MinimumSpanningTree,MST,KruskalAlgorithm,PrimAlgorithm,BoruvkaAlgorithm,
  MaximumFlow,MinimumCut,FordFulkersonAlgorithm,EdmondsKarpAlgorithm,DinicAlgorithm,
  BipartiteMatching,MaximumMatching,GraphColouring,MaximumClique,VertexCover,
  DegreeCentrality,BetweennessCentrality,ClosenessCentrality,EigenvectorCentrality,PageRank,
  CommunityDetection,LouvainAlgorithm,LabelPropagationAlgorithm,
  GraphPartitioning,GraphCut,RecursiveBisection,KernighanLinAlgorithm,
  GraphLaplacian,SpectralPartitioning,RandomWalk,TriangleCounting,
  TransitiveClosure,EulerianPath,ArticulationPoint,BridgeEdge,
  DynamicConnectivity,StreamingGraphAlgorithm,ParallelGraphAlgorithm,
  DistributedGraphAlgorithm,GPUGraphAlgorithm,Complexity,Verification,Reproducibility
 ]
 EXPAND(GraphAlgorithmFamilies) -> [
  TraversalAndReachabilityAlgorithms,
  ShortestPathAlgorithms,
  ConnectivityAndOrderingAlgorithms,
  SpanningTreeAlgorithms,
  FlowCutAndMatchingAlgorithms,
  ColouringCliqueAndCoverAlgorithms,
  CentralityCommunityAndSpectralAlgorithms,
  PartitioningDynamicAndParallelGraphAlgorithms
 ]
 STOP_CONDITION = "Stop when problem, preconditions, algorithm, representation, complexity, correctness and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Computational Graph Ontology"
 RELATED_ONTOLOGIES = [
  "Computational Graph Ontology",
  "Computational Data Structures",
  "Sparse Data Structure Ontology",
  "Computational Matrix Ontology",
  "Vector Computation Ontology",
  "Computational Complexity Ontology",
  "Algorithm Design Ontologies",
  "Sparse Matrix Ontology 2",
  "Numerical Linear Algebra Ontology 2",
  "High Performance Computing Ontology 2",
  "Parallel Computing Ontology",
  "Distributed Computing Ontology",
  "GPU Computing Ontology"
 ]
 ROUTE_RULES = [
  "Graph Algorithms -> Computational Graph for parent graph representation semantics",
  "Graph Algorithms -> Computational Data Structures for queues, stacks, heaps and union-find",
  "Graph Algorithms -> Sparse Data Structure for compressed adjacency representations",
  "Graph Algorithms -> Computational Matrix and Sparse Matrix for matrix-based graph methods",
  "Graph Algorithms -> Vector Computation for frontier, score and state vectors",
  "Graph Algorithms -> Computational Complexity and Algorithm Design for complexity and correctness",
  "Graph Algorithms -> Numerical Linear Algebra for spectral graph methods",
  "Graph Algorithms -> HPC, Parallel, Distributed and GPU Computing for scalable execution"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible graph algorithms, complexity models or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve graph-problem-versus-graph-representation distinction.",
  "Preserve algorithm-precondition-versus-output-validation distinction.",
  "Preserve traversal-order-versus-traversal-correctness distinction.",
  "Preserve exact-versus-approximate graph algorithm distinction.",
  "Preserve static-versus-dynamic graph algorithm distinction.",
  "Preserve serial-versus-parallel graph algorithm distinction.",
  "Preserve machine-readable graph-algorithm semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Graph algorithms are computational procedures for traversing, transforming and analysing graph structures."
 TRAVERSAL_SUMMARY = "BFS and DFS underpin reachability, connectivity and many derived graph procedures."
 PATH_SUMMARY = "Dijkstra, Bellman-Ford, Floyd-Warshall and A* solve different shortest-path regimes."
 STRUCTURE_SUMMARY = "MST, connectivity, topological ordering, flows, cuts and matching address core graph structure problems."
 ANALYTICS_SUMMARY = "Centrality, PageRank, communities, spectral methods and partitioning analyse large-scale graph organisation."
 VALIDATION_SUMMARY = "Graph algorithms should be checked against graph-type preconditions, complexity expectations and problem-specific invariants."
}

FLOW {
 INPUT
 -> detect_graph_algorithm_context
 -> identify_graph_type_direction_weights_size_density_problem_updates_memory_and_hardware
 -> classify_traversal_path_connectivity_MST_flow_matching_colouring_centrality_community_partitioning_or_dynamic_problem
 -> verify_algorithm_preconditions
 -> choose_algorithm_and_graph_representation
 -> configure_required_queue_stack_heap_union_find_frontier_or_matrix_support
 -> execute_algorithm
 -> analyse_time_space_work_communication_and_scalability
 -> if_large_choose_parallel_distributed_external_memory_or_GPU_variant
 -> verify_paths_components_trees_flows_cuts_orders_scores_or_partitions
 -> document_algorithm_parameters_representation_versions_and_provenance
 -> route_to_related_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-COMPUTATIONAL-DATA-STRUCTURES-GRAPH-ALGORITHMS-ONTOLOGY-V1"
}

}

END_FLOW