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Section 8.4. Connectivity. 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

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Section 8.4

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Section 8 4

Section 8.4

Connectivity


Paths

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 e1, … , en of the graph such that f(e1)={x0,x1}, f(e2)={x1,x2}, … , f(en)={x n-1,xn} where x0 = u and xn = v

  • In a simple graph, we denote this path by its vertex sequence


Paths1

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 x1, x2, …, xn-1

  • Simple path or circuit: path or circuit that does not contain the same edge more than once


Example 1

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


Example 11

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


Paths in directed multigraphs

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


Connectedness in undirected graphs

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


Connected components

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


Cut vertices and cut edges

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


Example

Example

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


Example1

Example

Original graph:

Vertex c is a cut vertex:

Vertex b is a cut vertex:

Vertex e is a cut vertex:


Example2

Example

Cut edges are:

{a, b}

{c, e}


Connectedness in digraphs

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


Examples

Examples

Strongly-connected

Weakly-connected


Paths and isomorphism

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


Section 8 41

Section 8.4

Connectivity


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