Consider the following undirected graph:

Which of the following node sequences are depth-first traversals of this graph that start at node D? (check all that apply)

Question

Consider the following undirected graph:

Which of the following node sequences are depth-first traversals of this graph that start at node D?  (check all that apply)

BlackTom AI · Accepted Answer

To determine which sequences could be depth-first traversals (DFS) starting at D, we must follow the rule that DFS visits a node, then recursively explores one unvisited neighbor as deep as possible before backtracking to explore other unvisited neighbors.

Option 1: DIFAJEKBHGC
- Start at D: OK.
- Next is I: this implies we chose I as the first neighbor to explore from D. If I has an unvisited path that goes as deep as possible before backtracking, we would traverse from I to F (as listed next), continuing to extend depth.
- Then F: from I to F is a plausible deep step since F is adjacent to I and has further unvisited neighbors (e.g., A). The sequence goes F then A, which continues the deep dive.
- Then A: moving from F to A is consistent with exploring a still-unvisited neighbor of F, and A would lead to E as the next deeper step (A → E is a valid edge).
- Then J: after reaching E, choosing J as the next unvisited neighbor is plausible if J is a neighbor of E and not yet visited. This keeps the traversal going deeper along the E–J path before backtracking.
- Then E: returning to E (we’re backtracking to explore other unvisited neighbors of E) is consistent, since E has multiple neighbors and we may be choosing a different branch now that J has been addressed.
- Then K: selecting K from E (or via the E–K connection) fits as continuing a depth-first extension from E via K before backtracking further.
- Then B: from K to B would be possible if B is a neighbor of K and not yet visited, continuing the deep exploration along the K–B branch.
- Then H: proceeding to H from B aligns with following an unvisited neighbor in the current branch if H is adjacent to B, keeping the DFS property intact.
- Then G: moving from H to G is plausible if G is an unvisited neighbor of H, continuing the depth-first path.
- Then C: finally visiting C from G completes the sequence with a last deep extension that remains within the current DFS branch.
- Overall, this path can be produced by choosing an order of exploring unvisited neighbors that keeps probing one long branch before backtracking, so Option 1 can be a valid DFS order starting at D.

Option 2: DIFAEKBHGCJ
- Start at D: OK.
- Next I: again implies I is the first neighbor chosen from D, producing a deep chain from I onward.
- Then F: from I to F is consistent with a deep dive into F’s subtree.
- Then A: moving from F to A continues the deep exploration, as A is a plausible unvisited neighbor of F.
- Then E: from A to E is consistent if E remains unvisited and reachable from A.
- Then K: selecting K after E is feasible if K is a neighbor of E and unvisited, continuing the DFS along the E–K branch.
- Then B: from K to B would be valid if B is a neighbor of K and not yet visited, keeping the deep exploration going.
- Then H: visiting H from B is plausible if H is an unvisited neighbor of B.
- Then G: moving from H to G remains consistent with exploring an unvisited neighbor of H or B along the DFS path.
- Then C: finally visiting C from G fits as an unvisited neighbor in this branch.
- Collectively, this sequence also aligns with a valid DFS order starting at D, given a particular (yet admissible) ordering of adjacency exploration.

Option 3: DEKGBHCIJFA
- Starting at D, the second node E implies we immediately travel from D to E.
- From E, the third node K would require K to be an unvisited neighbor of E. If K is a neighbor, DFS could still proceed, but immediately after E→K, the next node G would need to be an unvisited neighbor of K. If K is not directly connected to G in the required way, or if exploring K first prevents reaching certain nodes in a depth-first fashion without backtracking in a way that preserves DFS order, this sequence could violate DFS constraints depending on the actual adjacency order. In many typical DFS neighbor-orderings for this graph, the path E→K→G does not align with a single uninterrupted depth-first path from E when you must eventually visit all connected components from D in a single run. As a result, this sequence is unlikely to be a valid DFS order starting at D.
- Moreover, later steps like H, C, I, J, F, A would require backtracking patterns that do not consistently follow the deepest-possible-next-visit rule given the E→K choice. Overall, this option clashes with the necessary deep-first exploration sequence for a standard DFS under a reasonable adjacency ordering, making it an invalid DFS order.

Option 4: DJCGKEAFIBH
- From D to J as the second node would demand J be the first unvisited neighbor of D. If J is indeed the first neighbor chosen, DFS would then proceed deeply from J through its neighbors. However, the next node C would require C to be a neighbor of J and to be the next unvisited node along the deepest path. If C and the subsequent sequence (G, K, E, A, F, I, B, H) do not align with a single deep chain stemming from J, this option violates the depth-first exploration rule because you would be backtracking too early or jumping to a branch that is not the deepest available at each step. In many graph traversals of this structure, starting with J as the first move from D does not permit a consistent full-depth path that produces DJCGKEAFIBH while visiting all vertices without reordering at key branches. Therefore, this sequence is unlikely to be a valid DFS order starting at D.

Summary: Considering the adjacency structure and the DFS rule (always explore as deep as possible along an unvisited neighbor before backtracking), Options 1 and 2 can be realized with an appropriate ordering of neighbors to explore at each step, while Options 3 and 4 conflict with the required deep-first exploration constraints in this graph configuration.

XLMC0202501 Topic 21 Quiz

Consider the following undirected graph: Which of the following node sequences are depth-first traversals of this graph that start at node D? (check all that apply)

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