1. ساختمانهای گسسته دانشگاه صنعتی شاهرود – فروردین 1392

2. bipartite graph: (V,E) an undirected graph whose vertex set V can be partitioned in two disjoint, nonempty sets V 1 and V 2 such that every edge connects a vertex in V 1 to a vertex in V 2. V1V1 V2V2 V1V1 V2V2 2

3. Other Graph Problems: Assignment Problem Given: n people and n jobs Each person can be assigned to exactly one job Each job should be assigned to exactly one person Person-job compatibility is given by a directed graph (e.g., having a link A  x means “person A can do job x ”) Goal: Find an assignment of n jobs to n people (if such an assignment exists). Example: A CB people D E jobs xy zu v 3

4. 4 Trees A tree is a connected graph which contains no cycles. Properties of Tree Every tree with n vertices has exactly n  1 edges. Any two vertices in a tree are connected by exactly one path.

5. 5

6. Examples for Isomorphic Graphs

7. 7

8. 10 8 Planar Graphs and Graph Coloring  A graph is a planar graph if and only if it has a pictorial representation in a plane which is a plane graph. This pictorial representation of a planar graph G as a plane graph is called a planar representation of G.  Let G denote the plane graph in Figure 10.111. Graph G, in Figure 10.111, divides the plane into different regions, called the faces of G.

9. 10 9 Planar Graphs and Graph Coloring

10. 10

11. 11 Planar Graphs and Graph Coloring