Graphs

1. Graphs

2. Contents.. • Graph • Types of Graphs • Data Structures to Store Graphs • Graph Definitions

3. Graph • A graph is a pictorial representation of data. • A graph connects different vertices and provides a flow of data. Back

4. Types of Graphs • Definition (undirected, unweighted): • A Graph G, consists of • a set of vertices, V • a set of edges, E • where each edge is associated with a pair of vertices. • We write: G = (V, E)

5. Directed Graph: • Same as above, but where each edge is associated with an ordered pair of vertices.

6. Weighted Graph: • Same as above, but where each edge also has an associated real number with it, known as the edge weight. Back

7. Data Structures to Store Graphs • Adjacency Matrix Structure • Certain operations are slow using just an adjacency list because one does not have quick access to incident edges of a vertex. • We can add to the Adjacency List structure: • a list of each edge that is incident to a vertex stored at that vertex. • This gives the direct access to incident edges that speeds up many algorithms.

8. Adjacency Matrix • The standard adjacency matrix stores a matrix as a 2-D array with each slot in A[i][j] being a 1 if there is an edge from vertex i to vertex j, or storing a 0 otherwise. Back

9. Graph Definitions • A complete undirected unweighted graph • is one where there is an edge connecting all possible pairs of vertices in a graph. The complete graph with n vertices is denoted as Kn. • A graph is bipartite • if there exists a way to partition the set of vertices V, in the graph into two sets V1 and V2 • where V1 V2 = V and V1 V2 = , such that each edge in E contains one vertex from V1 and the other vertex from V2.

10. Complete bipartite graph • A complete bipartite graph on m and n vertices is denoted by Km,n and consists of m+n vertices, with each of the first m vertices is connected to all of the other n vertices, and no other vertices.

11. A weighted graph • A weighted graph associates a label (weight) with every edge in the graph. • The weight of a path or the weight of a tree in a weighted graph is the sum of the weights of the selected edges. • The function dist(v,w) • The function dist(v, w), where v and w are two vertices in a graph, is defined as the length of the shortest weight path from v to w. • dist(b,e) = 8

12. A subgraph • A graph G'= (V', E') is a subgraph of G = (V, E) if V'  V, E'  E, and for every edge e'  E', if e' is incident on v' and w', then both of these vertices are contained in V'.

13. A simple path • A simple path is one that contains no repeated vertices. • A cycle • A path of non-zero length from and to the same vertex with no repeated edges. • A simple cycle • A cycle with no repeated vertices except for the first and last ones. 6 5 4 1 3 2

14. A path • A path of length n from vertex v0 to vertex vn is an alternating sequence of n+1 vertices and n edges beginning with vertex v0 and ending with vertex vn in which edge eiincident upon vertices vi-1 and vi. • (The order in which these are connected matters for a path in a directed graph in the natural way.) • A connected graph • A connected graph is one where any pair of vertices in the graph is connected by at least one path Back

15. THANK YOU Back