graph-algorithms

# Graph Algorithms ## When to Use Use this skill when working on graph-algorithms problems in graph number theory. ## Decision Tree 1. **Traversal selection** - BFS: shortest paths (unweighted), level structure - DFS: cycle detection, topological sort, SCC 2. **Shortest path algorithms** | Algorithm | Use Case | Complexity | |-----------|----------|------------| | Dijkstra | Non-negative weights | O((V+E) log V) | | Bellman-Ford | Negative weights | O(VE) | | Floyd-Warshall | All pairs | O(V^3) | 3. **Minimum Spanning Tree** - Prim's: dense graphs, greedy from vertex - Kruskal's: sparse graphs, union-find - `z3_solve.py prove "cut_property"` 4. **Network Flow** - Max-flow = min-cut (Ford-Fulkerson) - Matching via flow network - `sympy_compute.py linsolve "flow_conservation"` 5. **Graph properties** - Spectral: eigenvalues of adjacency matrix - Connectivity: via DFS/BFS - Coloring: greedy or SAT reduction ## Tool Commands ### Sympy_Adjacency ```bash uv run python -m runtime.harness scripts/sympy_compute.py eigenvalues "adjacency_matrix" ``` ### Z3_Dijkstra ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "d[v] >= d[u] + w(u,v) for all edges" ``` ### Z3_Mst_Cut ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "min_edge_crossing_cut_in_mst" ``` ### Sympy_Flow ```bash uv run python -m runtime.harness scripts/sympy_compute.py linsolve "flow_conservation_equations" ``` ## Key Techniq...