Which of the following option is/are true?a)In a depth-first search of an undirected graph G, every edge of G is either a tree edge or a back edge.b)Forward and cross edges never occur in a depth-first search of an undirected graph.c)Post-order traversal in a binary tree is similar to depth first traversal.d)A directed graph is acyclic if and only if a depth-first search yields no back edges.Correct answer is option 'A,B,D'. Can you explain this answer?

Question

Answer

Option (A): TRUE

Depth search explores edge (u, v), it is in the direction from u to v, then vis undiscovered until that time, for otherwise, the search would have explored this edge already in the direction from v to u. Thus, (u, v) becomes a tree edge. If the search explores (u, v) first in the direction from v to u, then (u, v) is a back edge.

Option (B): TRUE

An undirected graph may entail some ambiguity in the classification of edges since (u, v) and (v, u) are really the same edge. Hence, forward and cross edges never occur in a depth-first search of an undirected graph.

Option (C): FALSE

Pre-order traversal in a binary tree is similar to a depth-first traversal.

Option (D): TRUE

Suppose that a depth-first search produces a back edge (u, v) Then vertex v is an ancestor of vertex u in the depth-first forest. Thus, G contains a path from v to u, and the back edge (u, v) completes a cycle.

Hence, the correct options are (A), (B) and (D).

Answer

Explanation:

a) In a depth-first search of an undirected graph G, every edge of G is either a tree edge or a back edge:
In a depth-first search (DFS) of an undirected graph, every edge of the graph is either a tree edge or a back edge.

- Tree edges: These are the edges that are discovered during the DFS traversal and form a spanning tree of the graph. They connect a vertex to one of its undiscovered neighbors.
- Back edges: These are the edges that connect a vertex to one of its ancestors in the DFS traversal. They form cycles in the graph.

Since an undirected graph does not have any distinction between incoming and outgoing edges, every edge encountered during the DFS will either be a tree edge or a back edge. Therefore, option a) is true.

b) Forward and cross edges never occur in a depth-first search of an undirected graph:
Forward and cross edges occur in a depth-first search of a directed graph, but not in an undirected graph.

- Forward edges: These are the edges that connect a vertex to one of its descendants in the DFS traversal. They go 'forward' in the direction of the traversal.
- Cross edges: These are the edges that connect two different branches of the DFS traversal. They do not go 'forward' or 'back' in the traversal.

In an undirected graph, there are no directions or descendants, so forward and cross edges cannot occur. Therefore, option b) is true.

c) Post-order traversal in a binary tree is similar to depth first traversal:
Post-order traversal in a binary tree is a type of depth-first traversal.

- Depth-first traversal: In a depth-first traversal, the algorithm explores as far as possible along each branch before backtracking. It can be done in three ways: pre-order, in-order, and post-order.
- Post-order traversal: In post-order traversal, the algorithm visits the left subtree, then the right subtree, and finally the root node.

Post-order traversal is similar to depth-first traversal because it explores the tree in a depth-first manner, but with a specific order of visiting the nodes. Therefore, option c) is true.

d) A directed graph is acyclic if and only if a depth-first search yields no back edges:
A depth-first search (DFS) of a directed graph can be used to determine if the graph contains any cycles. If the DFS yields no back edges, then the graph is acyclic.

- Back edges: These are the edges that connect a vertex to one of its ancestors in the DFS traversal. They form cycles in the graph.

If a DFS traversal encounters a back edge, it means that there is a cycle in the graph. Therefore, if a DFS yields no back edges, it implies that the graph is acyclic. Conversely, if a graph is acyclic, a DFS traversal will not encounter any back edges.

Therefore, option d) is true.

In summary, the correct options are A, B, and D:
- In a depth-first search of an undirected graph G, every edge of G is either a tree edge or a back edge.
- Forward and cross edges never occur in a depth-first search of an undirected graph.
- A directed graph is acyclic if and only if a depth-first search yields no back edges.

Similar Computer Science Engineering (CSE) Doubts

Let G be an undirected graph. Consider a depth-first traversal of G, and let T be the resulting depth-first search tree. Let u be a vertex in G and let v be the first new (unvisite d) vertex visited after visiting u in the traversal. Which of the following statement is always true?

Consider a DFS is implemented on an undirected weighted graph G. Let d(r,u) and d(r,v) be the weight of the edge (r,u) and edge (r, v) respectively. If v is visited immediately after u in depth first traversal, which of the following statement is correct?

Given a directed graph where weight of every edge is same, we can efficiently find shortest path from a given source to destination using?

Given a directed graph where weight of every edge is same, we can efficiently find shortest path from a given source to destination using?

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