If the DFS tree does not have any back edges, then there are no cycles in the graph. Group of answer choices True False

Understanding the Statement

The statement asserts that if a Depth-First Search (DFS) tree lacks back edges, it indicates the absence of cycles in the graph.

Relevant Concepts

1. DFS Tree: A tree formed by performing a depth-first search on a graph.
2. Back Edges: Edges that connect a vertex to one of its ancestors in the DFS tree.
3. Cycles in Graphs: A cycle exists when there is a path in the graph that begins and ends at the same vertex.

Analysis

1. In a directed or undirected graph, when conducting DFS, back edges play a crucial role in identifying cycles.
2. If a DFS tree does not have any back edges:
   • This means that there are no edges connecting a vertex to its ancestors.
   • Consequently, if there are no connections leading back to previously visited vertices, it implies that there are no cycles present.

Verify and Summarize

The logic supporting the statement holds true, as the absence of back edges directly leads to a tree structure that cannot recapitulate itself, thus confirming that no cycles exist.

Final Answer

True – If the DFS tree does not have any back edges, then there are no cycles in the graph.