MO412 - Mateus de Padua Vicente

Consider a directed graph  G = ( V , E )  and the Kosaraju-Sharir's algorithm for finding strongly connected components (SCCs). Analyze the following statements: I.   The SCC decomposition of a directed graph is unique, but the particular order in which SCCs are discovered by Kosaraju’s algorithm may depend on the order of adjacency lists. II.   If a directed graph has no back edges in a DFS traversal, then each vertex forms a distinct SCC. III.  If  G  is strongly connected, then the second DFS in Kosaraju’s algorithm produces a single tree covering all vertices, regardless of the order of exploration. IV.   The order of vertices given by decreasing finishing times of the first DFS guarantees that the second DFS on the transposed graph  G T  discovers one SCC at a time. Select the correct alternative: A) Only I and II are correct. B) Only I and IV are correct. C) Only I, II and IV are correct. D) Only I, III and IV are correct. E) ...

Consider a graph G = ( V , E ) G = (V, E) , and the Depth-First Search (DFS) and Breadth-First Search (BFS) algorithms. Analyze the following statements: I. In a DFS execution on a directed graph, the classification of edges as tree, forward, backward, and cross edges depends on the order in which vertices are explored. II. In a BFS executed from a source vertex s s  in an unweighted graph, the order in which vertices are discovered at the same level may vary depending on the order of edges in the adjacency list of the graph, but the minimum distance from each vertex to s s  remains unchanged. III. In directed graphs, a back edge detected during DFS indicates the presence of at least one cycle that includes the origin vertex of the edge. IV. In undirected graphs, BFS can detect all cycles of length 3 (triangles), but it does not guarantee detection of larger cycles. V. DFS guarantees that in a connected undirected graph, the tree generated contains exactly ∣ V ∣ − 1 |V| -...