1. Department of Computer Engineering Faculty of Engineering, Prince of Songkla University 1 10 - Graphs Original Slides by Rada Mihalcea

2. What is a Graph? A graph G = (V,E) is composed of: V: set of vertices E: set of edges connecting the vertices in V An edge e = (u,v) is a pair of vertices Example: a b c d e V= {a,b,c,d,e} E= {(a,b),(a,c),(a,d), (b,e),(c,d),(c,e), (d,e)}

3. Applications electronic circuits networks (roads, flights, communications) CS16 LAX JFK LAX DFW STL HNL FTL

4. Directed vs. Undirected Graph An undirected graph is one in which the pair of vertices in a edge is unordered, (v 0, v 1 ) = (v 1,v 0 ) A directed graph is one in which each edge is a directed pair of vertices, != tail head

5. Terminology: Adjacent and Incident If (v 0, v 1 ) is an edge in an undirected graph, v 0 and v 1 are adjacent The edge (v 0, v 1 ) is incident on vertices v 0 and v 1 If is an edge in a directed graph v 0 is adjacent to v 1, and v 1 is adjacent from v 0 The edge is incident on v 0 and v 1

6. 6 Terminology: Path path: sequence of vertices v 1,v 2,...v k such that consecutive vertices v i and v i+1 are adjacent. 3 3 3 3 2 a b c d e a b c d e a b e d cb e d c

7. More Terminology simple path: no repeated vertices cycle: simple path, except that the last vertex is the same as the first vertex a b c d e b e c

8. More… tree - connected graph without cycles forest - collection of trees

9. Graph Representations Adjacency Matrix Adjacency Lists

10. Adjacency Matrix Let G=(V,E) be a graph with n vertices. The adjacency matrix of G is a two-dimensional n by n array, say adj_mat If the edge (v i, v j ) is in E(G), adj_mat[i][j]=1 If there is no such edge in E(G), adj_mat[i][j]=0 The adjacency matrix for an undirected graph is symmetric; the adjacency matrix for a digraph need not be symmetric

11. Examples for Adjacency Matrix G1G1 G2G2 G4G4 0 12 3 0 1 2 1 0 2 3 4 5 6 7 symmetric undirected: n 2 /2 directed: n 2

12. Adjacency Lists (data structures) #define MAX_VERTICES 50 typedef struct node *node_pointer; typedef struct node { int vertex; struct node *link; }; node_pointer graph[MAX_VERTICES]; int n=0; /* vertices currently in use */ Each row in adjacency matrix is represented as an adjacency list.

13. 01230123 012012 0123456701234567 1 23 023 013 0 12 G1G1 1 0 2 G3G3 1 2 0 3 0 3 12 5 4 6 57 6 G4G4 0 12 3 0 1 2 1 0 2 3 4 5 6 7 An undirected graph with n vertices and e edges ==> n head nodes and 2e list nodes