The result of a depth-first search of a graph can be conveniently described in terms of a spanning tree of the vertices reached during the search. Based on this spanning tree, the edges of the original graph can be divided into three classes: forward edges, which point from a node of the tree to one of its descendants, back edges, which point from a node to one of its ancestors, and cross edges, which do neither. Sometimes tree edges, edges which belong to the spanning tree itself, are classified separately from forward edges. If the original graph is undirected then all of its edges are tree edges or back edges.
Question
The result of a depth-first search of a graph can be conveniently described in terms of a spanning tree of the vertices reached during the search. Based on this spanning tree, the edges of the original graph can be divided into three classes: forward edges, which point from a node of the tree to one of its descendants, back edges, which point from a node to one of its ancestors, and cross edges, which do neither. Sometimes tree edges, edges which belong to the spanning tree itself, are classified separately from forward edges. If the original graph is undirected then all of its edges are tree edges or back edges.
Solution
The above paragraph is describing the classification of edges in a graph based on a depth-first search (DFS). DFS is a method for traversing or searching tree or graph data structures. The algorithm starts at the root (selecting some arbitrary node as the root in the case of a graph) and explores as far as possible along each branch before backtracking.
In the context of a DFS of a graph, the edges can be classified into three types:
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Forward edges: These are the edges that point from a node of the tree to one of its descendants. In other words, these edges point towards the direction that the DFS is progressing in.
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Back edges: These are the edges that point from a node to one of its ancestors. These edges represent the situations where the DFS has to backtrack because it has reached a node that it has already visited.
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Cross edges: These are the edges that do not point to an ancestor or a descendant. These edges connect siblings or cousins in the tree.
Sometimes, a fourth category is also defined:
- Tree edges: These are the edges that belong to the spanning tree itself. These are the edges that the DFS uses to traverse the graph.
If the original graph is undirected, then all of its edges are either tree edges or back edges. This is because in an undirected graph, all edges are bidirectional, so the concept of forward and cross edges does not apply.
Similar Questions
The following six questions are about breadth first search and depth first search algorithms on a given graph. Consider an undirected graph 𝐺=(𝑉,𝐸)𝐺=(𝑉,𝐸) given below, where the set of vertices 𝑉={1, 2, 3, 4, 5}𝑉={1, 2, 3, 4, 5} and the set of edges 𝐸={(1, 2),(1, 3),(1, 4),(3, 2),(3, 5)}𝐸={(1, 2),(1, 3),(1, 4),(3, 2),(3, 5)}. We run BFS and DFS algorithms on this graph starting from the vertex 1 which is the initial vertex. Answer the following questions. 1 pointWhich are the three vertices that will be added to the BFS queue in the first step of the BFS algorithm? Does the order in which they are added matter? The three vertices will be 2, 3 and 4, their order will be exactly the same as the one given in this answer option. The three vertices will be 2, 3 and 4, their order does not matter. The three vertices will be 2, 3 and 5, their order will be exactly the same as the one given in this answer option. The three vertices will be 2, 3 and 5, their order does not matter.
It is an algorithm for traversing or searching tree or graph data structures.Question 3Select one:a.spanning treeb.breadth first search c.depth first search.
In relation to trees, what is the difference between a depth first and breadth first search? 1 pointA depth first will investigate nodes with greater detail, while a breadth first is more superficial in approach. A breadth first is more thorough so will return the result faster. A depth first approach will travel from top to bottom through sibling nodes, while a breadth first will travel through each level.
Consider the following undirected graph: A-B B-D D-F F-C A-C C-E C-B Which one of the following sequences is not a depth-first traversal of the above graph, when starting at node 'a'?
What is the role of Depth-First Search (DFS) in graph traversal?A) Ensures shortest path between nodesB) Visits nodes depth-wise until no more unvisited nodes are leftC) Calculates the average distance between nodesD) Finds the maximum flow in a graph
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