Answered: Show that if and only if…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Transcribed Image Text:**Depth-First Search (DFS) and Acyclic Graphs** --- **Statement:** Show that if and only if DFS (Depth-first Search) finds no back edges, the graph being traversed is acyclic. **Explanation:** When performing a Depth-first Search on a graph, edges encountered can be classified into four categories: 1. **Tree Edges:** These are edges that form part of the DFS tree. 2. **Back Edges:** These point back to an ancestor node in the DFS tree, indicating a cycle. 3. **Forward Edges:** These connect a node to a descendant node in the DFS tree, but are not part of the tree. 4. **Cross Edges:** These connect nodes in different branches. **Key Insight:** - A graph is acyclic if and only if the DFS traversal of the graph contains no back edges. This means that during the search, if no back edges are found, then there are no cycles in the graph, thus proving it to be acyclic. This statement provides a method to verify whether a given graph is acyclic using DFS. If during the DFS, every edge explored is a tree, forward, or cross edge, and no back edge is found, then the graph does not contain any cycles. Conversely, the presence of a back edge during DFS indicates a cycle. In summary, DFS is not only a tool for traversal but also a means of determining graph characteristics, such as acyclicity, by examining the types of edges found during the traversal process.

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