Discrete Structures CNS 2300 Text Discrete Mathematics and

Discrete Structures – CNS 2300 Text Discrete Mathematics and Its Applications (5 Edition) th Kenneth H. Rosen Chapter 8 Graphs

Section 8. 4 Connectivity

Paths l l 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 a e b g d f c

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

Paths in Directed Graphs a b c f e d

Acquaintanceship Graphs http: //www. cs. virginia. e du/oracle/ http: //www. brunching. c om/bacondegrees. ht ml Bacon 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 l 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 l Connected Undirected • Simple path between every pair of distinct vertices l Connected Directed • Strongly Connected • Weakly Connected

Euler & Hamilton Paths Bridges of Konigsberg

Euler Circuit l l 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 l l 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 l l 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 l l 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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