to Transparencies accompany Rosen, Discrete Mathematics and Its Applications
Section 9.4
Section 9.4 Connectivity We extent the notion of a path to undirected graphs. An informal definition (see the text for a formal definition): There is a path v0, v1, v2, . . . , vn from vertex v0 to vertex vn if there is a sequence of edges (joining the vertices in sequence) which can be followed from v0 to vn. The path has length n. The path is a circuit if the path begins and ends with the same vertex. A path is simple if it does not contain the same edge more than once. Note: There is nothing to prevent traversing an edge back and forth to produce arbitrarily long paths. This is usually not interesting which is why we define a simple path. _________________ Examples: Let G1 be the following graph:
Prepared by: David F. McAllister
TP 1
1999, 2007 McGraw-Hill
Transparencies to accompany Rosen, Discrete Mathematics and Its Applications
Section 9.4
u1
u2
u5 u4 There are many paths from u1 to u3 in G1:
u3
1) u1, u4, u2, u3; length = 3, the path is simple 2) u1, u5, u4, u1, u2, u3; length = 5, the path is simple and it contains a circuit u1, u5, u4, u1. 3) u1, u2, u5, u4, u3; length = 4, the path is simple How many simple paths are there?
Connectedness Definition: A simple graph is connected if there is a path between every pair of distinct vertices. ________________
Prepared by: David F. McAllister
TP 2
1999, 2007 McGraw-Hill
Transparencies to accompany Rosen, Discrete Mathematics and Its Applications
Section 9.4
Example: Let G be the following graph: v6
v7 v1 v5 v4
v8 v2 v3
The graph G2 is not connected since there is no path from v7 to v1. _____________________ Theorem: There is a simple path between every pair of distinct vertices in a connected graph. Proof: Because the graph is connected there is a path between u and v. Throw out all redundant circuits to make the path simple. _____________________
Prepared by: David F. McAllister
TP 3
1999, 2007 McGraw-Hill
Transparencies to accompany Rosen, Discrete Mathematics and Its Applications
Section 9.4
Example: In example 2) in the graph G1 above there is a circuit containing the vertex u1. Eliminate all edges in the path before the second occurrence of u1. _____________________ Definition: The maximally connected subgraphs of G are called the connected components or just the components. ______________________ Example: Let G be the following graph: v6
v7 v1 v5 v4
Prepared by: David F. McAllister TP 4
v8 v2 v3
1999, 2007 McGraw-Hill
Transparencies to accompany Rosen, Discrete Mathematics and Its Applications
Section 9.4
The components of the graph G are G1 = (V1 = {v6, v7, v8}, E1) and G2 = (V2 = {v1, v2, v3, v4, v5}, E2) where E1 and E2 contain all the edges which join the vertices in V1 and V2 respectively. ___________________ If one can remove a vertex (and all incident edges) and produce a graph with more components, the vertex is called a cut vertex or articulation point. Similarly if removal of an edge creates more components the edge is called a cut edge or bridge. ____________________ Examples: There are no cut edges or vertices in the graph G above. Removal of any vertex or edge does not create additional In components. the star network the center vertex is a cut vertex. All edges are cut edges.
Prepared by: David F. McAllister
TP 5
1999, 2007 McGraw-Hill
Transparencies to accompany Rosen, Discrete Mathematics and Its Applications
Section 9.4
K
1,8
In the following graphs G1 and G2 every edge is a cut edge. In the union, no edge is a cut edge. The vertex e is a cut vertex in all graphs. a e c G
1
b
a
f e
b
d c f g b e c g d
G2 d
a
GG 1 2
Prepared by: David F. McAllister
TP 6
1999, 2007 McGraw-Hill
Transparencies to accompany Rosen, Discrete Mathematics and Its Applications
Section 9.4
Connectedness in Directed Graphs Definition: A directed graph is strongly connected if there is (directed) path between every pair of vertices. If you can eliminate the arrows (turn the graph into an undirected one) and the graph is connected then the directed graph is weakly connected. _____________________ Examples: strongly connected (hence weakly connected)
not strongly connected but weakly connected.
Prepared by: David F. McAllister
TP 7
1999, 2007 McGraw-Hill
Transparencies to accompany Rosen, Discrete Mathematics and Its Applications
Section 9.4
_______________________ Paths and Isomorphism Isomorphic graphs must have 'isomorphic' paths. If one has a simple circuit if length k then so must the other. ________________ Theorem: Let M be the adjacency matrix for the graph G. Then the (i, j) entry of Mr is the number of paths of length r from vertex i to vertex j. Note: This is the standard power of M, not the boolean product. Proof below. First an example. __________________ Example: u1 u2
u5 u4
u3
Prepared by: David F. McAllister
TP 8
1999, 2007 McGraw-Hill
Transparencies to accompany Rosen, Discrete Mathematics and Its Applications
Section 9.4
0 1 M = 0 1 1 3 2 M 2 = 2 2 2 6 9 M 3 = 4 9 7
1 0 1 1 1 2 4 1 3 2 9 8 7 9 9
0 1 0 1 0 2 1 2 1 2 4 7 2 7 4
1 1 1 1 1 0 0 1...