1 Section 8.4 Connectivity

2 Paths In an undirected graph, a path of length n from u to v, where n is a positive integer, is a sequence of edges e 1, …, e n of the graph such that f(e 1 )={x 0,x 1 }, f(e 2 )={x 1,x 2 }, …, f(e n )={x n-1,x n } where x 0 = u and x n = v In a simple graph, we denote this path by its vertex sequence

3 Paths Circuit: a path that begins and ends at the same vertex (i.e., u=v) The path or circuit is said to pass through or traverse the vertices x 1, x 2, …, x n-1 Simple path or circuit: path or circuit that does not contain the same edge more than once

4 Example 1 In the simple graph at the left, the path: u1,u2,u4,u5 is a simple path of length 3 since {u1,u2}, {u2,u4}, {u4,u5} all exist as edges

5 Example 1 The path: u1,u2,u5,u4 is not a path, because no edge exists between u2 and u5 The path: u1,u2,u6,u5,u1 is a circuit of length 4

6 Paths in directed multigraphs Definition is virtually identical to definition of simple graph path When no multiple edges exist, the graph is denoted by its vertex sequence A circuit or cycle is a path that begins and ends at the same vertex A path or circuit is simple if it does not contain the same edge more than once

7 Connectedness in undirected graphs An undirected graph is connected if there is a path between every pair of distinct vertices The graph on the right is connected - can find at least one path between every pair of vertices

8 Connected components A graph that is not connected is the union of two or more connected subgraphs, each pair of which has no vertex in common These disjoint connected subgraphs are the connected components of the graph

9 Cut vertices and cut edges A cut vertex (or articulation point) is a vertex which, when removed with all its incident edges, leaves behind a subgraph with more connected components than were found in the original graph The removal of a cut vertex from a connected graph produces a subgraph that is not connected An edge whose removal produces a graph with more connected components than in the original graph is called a cut edge or bridge

10 Example Find the cut vertices and cut edges in the graph below:

11 Example Original graph: Vertex b is a cut vertex: Vertex c is a cut vertex: Vertex e is a cut vertex:

12 Example Cut edges are: {a, b}{c, e}

13 Connectedness in digraphs Strongly connected: a digraph is strongly connected if, for vertices a and b, there is a path from a to b and a pathfrom b to a Weakly connected: a digraph is weakly connected if there is a path between any two vertices in the underlying undirected graph A strongly connected graph is also weakly connected, but a weakly connected graph may not be strongly connected

14 Examples Strongly-connectedWeakly-connected

15 Paths and Isomorphism The existence of a simple circuit of length k, where k > 2, is a useful isomorphic invariant for simple graphs If one graph has such a circuit and the other does not, the graphs are not isomorphic

16 Section 8.4 Connectivity