gt-graph-traversal-orderings

name	gt - Graph Traversal Orderings
description	Orderings derived from graph traversal algorithms — BFS for breadth, DFS for depth, topological for dependencies.
output	{"format":"prose"}

Graph Traversal Orderings

Input: $ARGUMENTS

---

Overview

Many problems have graph structure (dependencies, relationships, hierarchies). The traversal order determines what you find and how efficiently. BFS explores breadth-first (all neighbors before going deeper), DFS explores depth-first (follow one path to its end), topological ordering respects dependencies.

Core Principle

Structure determines traversal. Dependencies need topological order. Shortest paths need BFS. Full exploration needs DFS. Match the algorithm to what you need.

Ordering Rules

Rule 1: BFS — Breadth First

• Explore all immediate neighbors before going deeper
• Guarantees shortest path in unweighted graphs
• When: finding shortest/quickest route, exploring all options at each level, level-order processing

Rule 2: DFS — Depth First

• Follow one path as far as it goes before backtracking
• Uses less memory, finds paths quickly (not necessarily shortest)
• When: exhaustive search, maze solving, detecting cycles, topological sort

Rule 3: Topological — Dependencies First

• Process nodes only after all their prerequisites are processed
• Only works on directed acyclic graphs (DAGs)
• When: build systems, task scheduling, prerequisite chains

Rule 4: Dijkstra/A* — Cost-Guided

• Expand the cheapest unexplored node (Dijkstra) or cheapest + heuristic (A*)
• Guarantees optimal paths in weighted graphs
• When: finding cheapest/fastest route with varying costs

Rule 5: Reverse Postorder — Dependencies via DFS

• DFS but process nodes in reverse order of completion
• Produces topological ordering
• When: compiler optimizations, strongly connected components

Application Procedure

Step 1: Model as Graph

• What are the nodes? (tasks, states, decisions, entities)
• What are the edges? (dependencies, transitions, relationships)
• Directed or undirected? Weighted or unweighted? Cyclic?

Step 2: Select Traversal

• Need shortest path (unweighted) → BFS
• Need exhaustive exploration → DFS
• Need dependency ordering → Topological
• Need cheapest path (weighted) → Dijkstra/A*

Step 3: Execute and Track

• Mark visited nodes to avoid cycles
• Record the traversal order
• Extract the answer from the traversal

When to Use

• Task scheduling with dependencies
• Exploring decision spaces
• Finding paths or connections
• Any problem with graph/tree structure

Verification

• Problem modeled as graph correctly
• Traversal algorithm matched to need
• All reachable nodes visited (or search terminated correctly)
• Dependencies respected (no node processed before prerequisites)