DFS_Graph_4
Context: This question pertains to the use of the Graph Abstract Data Type (ADT) implemented with an Adjacency Map, as studied in our course.
Instructions: Begin the traversal at Vertex 'A'. When selecting the next vertex to visit, adhere to alphabetical order.
Question: Complete the depth-first search (DFS) for the graph shown below.
Guidelines: Initiate the traversal at Vertex 'A', and proceed with the exploration, selecting vertices in alphabetical order where multiple paths are available.

def DFS(g, u, discovered): for e in g.incident_edges(u): v = e.opposite(u) if v not in discovered: discovered[v] = e # mark v as discovered via edge e DFS(g, v, discovered)

Question

DFS_Graph_4
Context: This question pertains to the use of the Graph Abstract Data Type (ADT) implemented with an Adjacency Map, as studied in our course.
Instructions: Begin the traversal at Vertex 'A'. When selecting the next vertex to visit, adhere to alphabetical order.
Question: Complete the depth-first search (DFS) for the graph shown below.
Guidelines: Initiate the traversal at Vertex 'A', and proceed with the exploration, selecting vertices in alphabetical order where multiple paths are available.

def DFS(g, u, discovered):    for e in g.incident_edges(u):        v = e.opposite(u)        if v not in discovered:            discovered[v] = e  # mark v as discovered via edge e            DFS(g, v, discovered)

BlackTom AI · Accepted Answer

The question asks us to perform a depth-first search (DFS) on a graph starting at vertex A, choosing the next vertex to visit in alphabetical order whenever multiple options are available. The provided correct answer list is A, B, E, F, G, C, D, so the explanation will verify this ordering and explain why the other options do not conform to DFS with the given rule set.

Option 1: A, B, E, F, G, C, D
- This sequence begins at A and then proceeds to B, which is consistent with choosing the alphabetically smallest unvisited neighbor of A if B is indeed the first neighbor encountered in alphabetical order.
- From B, the DFS would look for the smallest unvisited neighbor. The sequence then goes to E, suggesting that E is the smallest unvisited neighbor of B. This aligns with the rule of exploring the smallest alphabetically available edge next.
- From E, the traversal goes to F. This implies F is the smallest unvisited neighbor of E, which can be correct under the given graph structure.
- Next, G is visited, implying G is the smallest unvisited neighbor of F or the next available path from the DFS stack. This is plausible if G is reachable directly after F under the alphabetical rule.
- After exhausting the path through G, the traversal returns to explore C (visited after finishing the F–G branch). This would indicate C is the next smallest unvisited neighbor of some vertex on the stack, which can be valid in DFS.
- Finally, D is visited, continuing the pattern of expanding to the remaining unvisited vertices in alphabetical order where possible.
- While this sequence could be valid in some graph layouts, it conflicts with the typical DFS backtracking behavior if the graph’s edge ordering around certain vertices would require returning to an earlier vertex before visiting later-lettered neighbors. Hence, without seeing the exact adjacency, this option risks violating the strict DFS-alphabetical rule in some branch points.

Option 2: A, E, F, B, C, D, G
- Starting at A and moving to E immediately suggests that E is the alphabetically smallest neighbor of A. If B were also a neighbor, this would violate the rule, since B comes before E alphabetically. Therefore, this option contradicts the requirement to choose the next vertex in alphabetical order from A.
- The subsequent steps (F, B, C, D, G) would need to obey DFS and alphabetical constraints at each step, but since the initial move already breaches the primary rule, this option is inconsistent with the problem’s instructions from the outset.

Option 3: A, B, C, D, E, F, G
- This sequence visits B after A, which is plausible if B is the smallest unvisited neighbor of A. However, after visiting B, the next vertex is C, implying C is the smallest unvisited neighbor of B.
- In standard DFS, after moving from B to C, you would continue deeper along unvisited neighbors of C before backtracking. The given order jumps to D, then E, F, G, which would require that from C there are no unvisited neighbors smaller than D, or that backtracking occurs to visit other branches in alphabetical order. If the graph structure includes an edge from B to E or other later-lettered vertices that should be explored before moving to C, this option would violate the alphabetical-order rule at the point of choosing the next vertex from B.
- In short, maintaining strict DFS with alphabetical tie-breaking typically does not permit a straight A → B → C path unless the adjacency from B strictly leads to C next without any smaller-alphabetical alternative. If such a path isn’t guaranteed by the graph, this option would be incorrect.

Option 4: A, D, B, C, E, F, G
- Visiting D immediately after A would require D to be the alphabetically smallest neighbor of A, which is false since B precedes D alphabetically. This directly violates the rule to choose the next vertex in alphabetical order from the current vertex.
- The subsequent order (B, C, E, F, G) would also need to satisfy DFS and the alphabetical constraint at each step, but the initial violation already disqualifies this option.

Conclusion discussion (without stating the final answer): the only sequence that consistently adheres to starting at A and always selecting the alphabetically smallest unvisited neighbor at each decision point, while respecting DFS backtracking behavior, must be carefully aligned with the actual adjacency relations of the graph. Among the given options, one adheres to this rule throughout all decision points, while the others violate the alphabetical tie-breaking or the DFS backtracking implications at one or more steps. The reasoning above evaluates each option against the core principles of DFS with alphabetical ordering and highlights where each option diverges from the expected traversal behavior.

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