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Chapter 9: Graphs

Mark Allen Weiss: Data Structures and Algorithm Analysis in Java. Chapter 9: Graphs. Applications of Depth-First Search. Lydia Sinapova, Simpson College. Graph Connectivity. Connectivity Biconnectivity Articulation Points and Bridges Connectivity in Directed Graphs. Connectivity.

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Chapter 9: Graphs

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  1. Mark Allen Weiss: Data Structures and Algorithm Analysis in Java Chapter 9: Graphs Applications of Depth-First Search Lydia Sinapova, Simpson College

  2. Graph Connectivity • Connectivity • Biconnectivity • Articulation Points and Bridges • Connectivity in Directed Graphs

  3. Connectivity Definition: An undirected graph is said to be connected if for any pair of nodes of the graph, the two nodes are reachable from one another (i.e. there is a path between them). If starting from any vertex we can visit all other vertices, then the graph is connected

  4. A B C D E A B C D E Biconnectivity A graph is biconnected, if there are no vertices whose removal will disconnect the graph. not biconnected biconnected

  5. A B C D E Articulation Points Definition: A vertex whose removal makes the graph disconnected is called an articulation point or cut-vertex Cis an articulation point We can compute articulation points using depth-first search and a special numbering of the vertices in the order of their visiting.

  6. A B C D E Bridges Definition: An edge in a graph is called a bridge, if its removal disconnects the graph. Any edge in a graph, that does not lie on a cycle, is a bridge. Obviously, a bridge has at least one articulation point at its end, however an articulation point is not necessarily linked in a bridge. (C,D) and (E,D) are bridges

  7. A B C D E F Example 1 C is an articulation point, there are no bridges

  8. A B C D E F Example 2 Cis an articulation point, CB is a bridge

  9. D A G B C F E Example 3 B and C are articulation points, BC is a bridge

  10. A B C D E Example 4 B and C are articulation points. All edges are bridges

  11. A B C D E F G Example 5 Biconnected graph - no articulation points and no bridges

  12. A B C D E Connectivity in Directed Graphs (I) Definition: A directed graph is said to be strongly connected if for any pair of nodes there is a path from each one to the other

  13. A B C D E Connectivity in Directed Graphs (II) Definition: A directed graph is said to be unilaterally connected if for any pair of nodes at least one of the nodes is reachable from the other Each strongly connected graph is also unilaterally connected.

  14. A B C D E Connectivity in Directed Graphs (III) Definition: A directed graph is said to be weakly connected if the underlying undirected graph is connected Each unilaterally connected graph is also weakly connected There is no path between B and D

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