Discrete Structures – CNS2300 Text Discrete Mathematics and Its Applications (5th Edition) Kenneth H. Rosen Chapter 8 Graphs

Section 8.4 Connectivity

Paths A path is a sequence of edges that begins at a vertex of a graph and travels along edges of the graph, always connecting pairs of adjacent vertices. The path is a circuit if it begins and ends at the same vertex. The path or circuit is said to pass through the vertices or traverse the edges A path or circuit is simple if it does not contain the same edge more than once.

Paths a,b ,d ,g ,f e g a b d f c

Circuits, Simple Path or Circuit b c d e f g

Paths in Directed Graphs b c d e f

Acquaintanceship Graphs http://www.cs.virginia.edu/oracle/ http://www.brunching.com/bacondegrees.html Bacon No. No. People 0 1 2 3 4 4 6 7 8 9 10 1 1479 115204 285929 65021 4535 534 81 28 1 1

Counting Paths Between Vertices Let G be a graph with adjacency matrix A. The number of different paths of length r from vi to vj, where r is a positive integer, equals the (i, j)th entry of Ar

Connectedness Connected Undirected Connected Directed Simple path between every pair of distinct vertices Connected Directed Strongly Connected Weakly Connected

Euler & Hamilton Paths Bridges of Konigsberg

Euler Circuit An Euler circuit in a graph G is a simple circuit containing every edge of G. An Euler path in G is a simple path containing every edge of G.

Necessary & Sufficient Conditions A connected multigraph has an Euler circuit if and only if each of its vertices has even degree A connected multigraph has an Euler path but not an Euler circuit if and only if it has exactly two vertices of odd degree.

Hamilton Paths and Circuits A Hamilton circuit in a graph G is a simple circuit passing through every vertex of G, exactly once. An Hamilton Path in G is a simple path passing through every vertex of G, exactly once.

Conditions If G is a simple graph with n vertices n>=3 such that the degree of every vertex in G is at least n/2, then G has a Hamilton circuit. If G is a simple graph with n vertices n>=3 such that deg(u)+deg(v)>=n for every pair of nonadjacent vertices u and v in G, then G has a Hamilton circuit.

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