1. Introduction to Graph Theory Lecture 17: Graph Searching Algorithms

2. Introduction DFS forest allows us to identify edges as tree or back edges. With the DFS forest, we are able to solve numerous graph-processing problems. We’ll discuss few simple ones in this lecture  Cycle detection  Two-way Euler tour  Spanning tree  Two-colorability

3. Cycle Detection Look for the existence of back/down edge. A back/down edge v-w belongs to a cycle consisting of the edge and the tree path connecting the two nodes. A graph is acyclic if and only if no back/down edges.

4. Two-way Euler Tour Finding a walk that uses all the edges in the graph exactly twice. Go back and forth on each back link (or down link, but not both)

5. Spanning Tree This is exactly what DFS generates  V(G)-1 recursive calls, one for each edge on a spanning tree.

6. Two-Colorability Two-colorrability (bipartiteness) = no-odd-cycle  Assigning alternate colors as we proceed down the DFS tree  Check back edge for consistency in the coloring.  Any back edge connecting two vertices of the same color => odd cycle

7. Separability We’re interested in finding bridges  Review: a bridge is an edge that, if removed, would turn a connected graph to disconnected. In any DFS tree, a tree edge v-w is a bridge if and only if there are no back edges that connects a descendant of w to an ancestor of w.  We also consider w its own descendant (so edge 6-2 is not a bridge in the next example) Breaking the v-w edge would disconnect the sub-tree rooted w from the rest of the graph.

8. (cont) Lowest preorder number referenced by any back edge in the subtree rooted at the vertex

9. Biconnectivity A graph is biconnected if  It has no cut-vertex  There exist two disjoint paths between any pair of vertices. Question: How do we determine if a graph is biconnected using a DFS-based approach?

10. (cont) After brainstorming, we reached the following conclusion:  In any DFS tree, a vertex w is a cut-vertex if and only if, for every subtree rooted at vertex u with st[u]=w, there is a back edge that connects a descendant of w to an ancestor of w.  Need to check if the root of the tree is an articulation point (what would be the criterion for that?)

11. Breadth-First-Search If what we want is to find the shortest path, DFS is no use here. With BFS, we need a FIFO queue (instead of a stack for DFS)  Taking edges from the queue until finding one that points to an unvisited vertex  Visit that vertex by putting onto the queue all edges that go from that vertex to unvisited vertices.

12. Example Level 1 Level 2 Level 3

13. Code

14. Applications of BFS Shortest path Single-source shortest path All-pairs shortest path