This trace follows the actual state transitions behind the companion Representing a Graph Before Traversing It. It describes a common execution path; implementation details can vary, so keep the contract separate from the mechanism.
Step 1: Build the neighbor representation
A graph models entities as vertices and relationships as edges. Before choosing an algorithm, decide whether edges are directed, weighted, duplicated, or allowed to change. Those choices affect both the meaning of a path and the representation needed to answer queries efficiently.
Step 2: Initialize the frontier
An adjacency list stores each vertex with its outgoing neighbors, so scanning all neighbors costs in proportion to the degree. Breadth-first search uses a queue and discovers vertices in increasing edge distance for an unweighted graph. Depth-first search follows one branch until it must backtrack, which is useful for reachability and cycle reasoning.
Step 3: Expand one vertex
In breadth-first search, mark a neighbor when it enters the queue so a second edge cannot enqueue it again; in weighted graphs, choose an algorithm whose assumptions match the edge weights.
At this point, record the state that changed and check the invariant before advancing. If the operation repeats, make clear which values persist and which are recomputed.
Step 4: Mark each discovery once
An adjacency matrix uses quadratic space but makes edge existence checks constant time and can be effective for dense graphs. BFS does not find a minimum-cost path when edge weights differ; Dijkstra requires nonnegative weights, while negative edges need other methods. Marking a node visited at the right time prevents duplicate queue growth.
Step 5: Return reachability or distance
For a directed graph with an unreachable component, describe how you would count components and why one traversal from a single start node is insufficient.
The trace is complete when the result satisfies the stated contract. Compare this model with the concrete runtime or system you are studying before making a performance claim.