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Is following statement true/false? A DFS of a directed graph always produces the same number of tree edges, i.e., independent of the order in which vertices are considered for DFS.
- a)True
- b)False
Correct answer is option 'B'. Can you explain this answer?
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Is following statement true/false? A DFS of a directed graph always pr...
Consider the following graph. If we start from 'a', then there is one tree edge. If we start from 'b', then there is no tree edge. a→b
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Is following statement true/false? A DFS of a directed graph always pr...
Explanation:
Depth-first search (DFS) is a graph traversal algorithm that starts traversing the graph from a source vertex and explores as far as possible along each branch before backtracking. During the traversal, DFS maintains a tree structure that consists of tree edges, back edges, forward edges, and cross edges.
- Tree edges: Edges that are part of the DFS tree.
- Back edges: Edges that connect a vertex to an ancestor in the DFS tree.
- Forward edges: Edges that connect a vertex to a descendant in the DFS tree.
- Cross edges: Edges that connect vertices that are neither ancestors nor descendants of each other in the DFS tree.
The number of tree edges in a DFS of a directed graph depends on the order in which vertices are considered for DFS. Therefore, the given statement is false.
Example:
Consider the following directed graph:
```
A -> B -> C
| | |
v v v
D -> E -> F
```
If we start DFS from vertex A, we get the following DFS tree:
```
A -> B -> C
| | |
v v v
D E F
```
The DFS contains 5 tree edges.
If we start DFS from vertex D, we get the following DFS tree:
```
D -> E -> F
| | |
v v v
A -> B -> C
```
The DFS contains 4 tree edges.
Therefore, the number of tree edges in a DFS of a directed graph depends on the order in which vertices are considered for DFS.
Depth-first search (DFS) is a graph traversal algorithm that starts traversing the graph from a source vertex and explores as far as possible along each branch before backtracking. During the traversal, DFS maintains a tree structure that consists of tree edges, back edges, forward edges, and cross edges.
- Tree edges: Edges that are part of the DFS tree.
- Back edges: Edges that connect a vertex to an ancestor in the DFS tree.
- Forward edges: Edges that connect a vertex to a descendant in the DFS tree.
- Cross edges: Edges that connect vertices that are neither ancestors nor descendants of each other in the DFS tree.
The number of tree edges in a DFS of a directed graph depends on the order in which vertices are considered for DFS. Therefore, the given statement is false.
Example:
Consider the following directed graph:
```
A -> B -> C
| | |
v v v
D -> E -> F
```
If we start DFS from vertex A, we get the following DFS tree:
```
A -> B -> C
| | |
v v v
D E F
```
The DFS contains 5 tree edges.
If we start DFS from vertex D, we get the following DFS tree:
```
D -> E -> F
| | |
v v v
A -> B -> C
```
The DFS contains 4 tree edges.
Therefore, the number of tree edges in a DFS of a directed graph depends on the order in which vertices are considered for DFS.
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