1. CSNB143 – Discrete Structure Topic 9 – Graph

2. Learning Outcomes Student should be able to identify graphs and its components. Students should know how to use certain ways to solve problems in graphs.

3. Topic 9 – Graph Basic Components of Graph A graph G consist of a finite set V of objects called vertices, a finite set E of objects called edges and a function , that assigns two vertices to each edge. We will write graph as G = (V, E,  ). If e is an edge, and  (e) = {v, w), that are vertices for e, we say e is an edge between v and w. The vertices v and w are called the end points of e. Usually, graphs are represented by pictures, using a point for each vertex and a line for each edge, as below:

4. Topic 9 – Graph Basic Components of Graph The degree of a vertex is the number of edges connected to it. Any edge that connects from a vertex to the same vertex, such as loop, contributes 2 to the degree of a vertex. A vertex with degree 0 is called an isolated vertex. A pair of vertices that determine an edge, or sharing the same edge, are adjacent vertices. Example Vertex A has degree 2, vertex B has degree 4, vertex C has degree 1, vertex D has degree 3, vertex E has degree 2.

5. Topic 9 – Graph Basic Components of Graph Work this out Find the degrees for all vertices in (b) and (c). Can you identify any isolated vertex, adjacent vertices and non-adjacent vertices in (b)?

6. Topic 9 – Graph Basic Components of Graph A path  in a graph G consists of a vertex sequence of  : v 1, v 2, …, v k, each one is adjacent to the next, and no edge occurs more than once. In short, path is a journey from vertex v 1 and travel through the edges to v k without using any edge twice. A circuit is a path that begins and ends at the same vertex. (circuit = cycle). A path is called simple if no vertex appears more than once in the vertex sequence.

7. Topic 9 – Graph Basic Components of Graph In (a), path  1 : A, B, E, D, D; path  2 : D, E, B, C and path  3 : A, B, A. Note that in  3, we do not know which path between A and B has been used first. In (b), path  4 : a, b, c, a; and  5 : d, c, a, a, b. Path  4 is a cycle. Path  6 : c, a, b, c, d is not simple. In (c), sequence 1, 2, 3, 2 is not a path, because the single edge between 2 and 3 has been traveled twice. Path  7 : 4, 5, 6, 4 is a cycle.

8. Topic 9 – Graph Basic Components of Graph A graph is called connected if there is a path from any vertex to any vertex in the graph. If not, the graph is called disconnected. If the graph is disconnected, the various connected pieces are called the components of the graph. Graph (a) is connected. Graphs (b) and (c) are disconnected graphs. Graph (c) has two components Graph (a) is connected. Graphs (b) and (c) are disconnected graphs. Graph (c) has two components.

9. Topic 9 – Graph Sub graph Let say G = (V, E,  ) is a graph. Choose a subset E 1 from E and subset V 1 from V so that V 1 contains all the end points of edges in E 1. Then H = (V 1, E 1,  1 ) is also a graph where  1 is  restricted to edges in E 1. H is called the sub graph of G. Example: (d), (e) and (f) are the sub graphs of (a).

10. Topic 9 – Graph Euler Path and Circuit A path in a graph G is called a Euler path if it includes EVERY EDGE exactly once. A Euler circuit is a Euler path that is a circuit. Euler path exists in (a); E, D, B, A, C, D, but Euler circuit does not exist. Euler circuit exist in (b); 5, 3, 2, 1, 3, 4, 5.

11. Topic 9 – Graph Euler Path and Circuit Example: Consider the floor plan as below. Each room is connected to every room and to the outside. Is it possible to begin in a room or outside and take a walk that goes through each door exactly once?

12. Topic 9 – Graph Euler Path and Circuit When we are looking at a graph, is it possible to determine whether a Euler path or Euler circuit exists without actually finding it? Theorem 1 (to identify a circuit) If a graph G has a vertex of odd degree, there can be no Euler circuit in G. If G is a connected graph and every vertex has even degree, then there is a Euler circuit in G.

13. Topic 9 – Graph Euler Path and Circuit Theorem 2 (to identify a path) If a graph G has more than two vertices of odd degree, then there can be no Euler path in G. If G is connected and has exactly two vertices of odd degree, there is a Euler path in G. The path must begin at one vertex of odd degree and end at the other.

14. Topic 9 – Graph

15. Euler Path and Circuit So the previous graph will be: Degrees for the four vertices are: A = 4, B = 4, C = 5, D = 7. From Theorem 1, there is odd degree vertex, so, there is no Euler circuit. From Theorem 2, there are exactly two odd degree vertices, so there is a Euler path. It must begin from C/D and ends at D/C. The path is: C, D, C, A, D, A, B, D, B, C, D

16. Topic 9 – Graph Euler Path and Circuit Work this out: Consider the graphs below, and find a Euler path or circuit.

17. Topic 9 – Graph Hamilton Path and Circuit A Hamilton path is a path that contains EACH VERTEX exactly once. Hamilton circuit is a circuit that contains each vertex exactly once except for the first vertex, which is also the last. Example: In (a), path c, d, e, b, a is a Hamilton path because it contains each vertex exactly once. There is no Hamilton circuit.

18. Topic 9 – Graph Hamilton Path and Circuit In (b), there is no Hamilton path or circuit because it is a non connected graph. In (c), path A, D, C, B, A is a Hamilton circuit. In (d), there is no Hamilton path or Hamilton circuit.

19. Topic 9 – Graph Hamilton Path and Circuit When we saw a graph, is it possible to determine whether a Hamilton path or circuit exists or not? (WARNING: They don’t necessarily work !!) Theorem 1: Let G be a connected graph with n vertices, n > 2 and no loops or multiple edges, G has a Hamilton circuit if for any two vertices u and v of G that are not adjacent, the degree of u plus the degree of v is greater that or equal to n. D(u) + D(v)  the number of vertices. Corollary 1 (outcome): G has a Hamilton circuit if each vertex has degree greater than or equal to n/2

20. Topic 9 – Graph Hamilton Path and Circuit Theorem 2: Let the number of edges of G be m. Then G has a Hamilton circuit if m  ½ (n 2 – 3n + 6) where n is the number of vertices. Reminder: If Theorem failed, we cannot say that the graph has no Hamilton circuit. We have to check the circuit or path manually.

21. Topic 9 – Graph Hamilton Path and Circuit Example: Take two non adjacent vertices; let say A and E. The sum of their degrees are (2 + 2) = 4 < 8 (number of vertices). But if we check properly in manual, we can have the Hamilton circuit that is A, B, C, D, E, F, G, H, A

22. Topic 9 – Graph Work this out Determine if the following graph has the Hamiltonian circuit and/or path. Discuss your answer