4.1 Introduction to Graphs

1. 4.1 Introduction to Graphs • Graph – consists of vertices and edges. Each edge must start and end at a vertex. • Example:

2. 4.1 Introduction to Graphs • Examples of non-graphs:(edge splits in two pieces, no vertex at end)

3. 4.1 Introduction to Graphs • Path – connects one vertex to another by a sequence of edges • Example: V1- V3- V2- V4 V1 V2 V3 V4 V5

4. 4.1 Introduction to Graphs • Connected Graph – there is a path from every vertex to every other vertex • Example of graph that is not connected: V1 V2 V6 V3 V4 V5

5. 4.1 Introduction to Graphs • Degree of a vertex - number of edges connected to it • Example: V4 has degree = 3 V1 V2 V3 V4 V5

6. 4.1 Introduction to Graphs • Total degree of a graph - sum of the degrees of all the vertices. Note: If a graph has n edges, the total degree = 2n. • Example: graph has total degree = 10 V2 V1 V4 V3

7. 4.2 Edge Paths • Konigsberg Bridge Problem: Can you walk across each bridge exactly once? North Bank River Island Island South Bank

8. 4.2 Edge Paths • In 1736, Euler showed that the Konigsberg bridge problem was unsolvable by using graph theory. • Another example – trace without lifting your pencil:

9. 4.2 Edge Paths • Edge Path – a path that goes through each edge of the graph exactly onceTo determine if a graph has an edge path, count the number of vertices of odd degree. Note that this number will always be even (0,2,4,…) • A graph has an edge path if it has: • 2 vertices of odd degree

10. 4.2 Edge Paths • Selecting the starting/ending vertices:Given 2 vertices of odd degree – start at one of the vertices and end at the other

11. 4.2 Fleury’s Method for Finding an Edge Path • Select a starting vertex. • Select and number a connecting edge if it is the only remaining unused edge, or, select it if the edge does not cut off or isolate a portion of the graph. • Repeat step 2 until all edges have been selected.

12. 4.3 Edge Cycles • Cycle – a path that starts and ends on the same vertex with no repeated edges • Edge Cycle – a cycle that includes every edge in the graph • Theorem: A graph has an edge cycle  it is connected and every vertex is of even degree

13. 4.3 Edge cycles • Selecting the starting/ending vertices for edge cycles: • no vertices of odd degree – start at any vertex and end at the same vertex

14. 4.3 Edge Cycles • Edge cycles are different from edge paths: • Edge Cycle – every vertex is of even degree • Edge Path – exactly 2 vertices are of odd degree

15. 4.3 Edge Cycles • Eulerizing – adding edges where they already exist to create an edge cycle • Method – add edges to connect the vertices of odd degree. This changes both vertices to an even degree. Make sure you only connect one edge to each odd vertex! • Note: Eulerizing is useful in museums, gardens, street systems with non-Euler cycles that force you to retrace your path

16. 4.4 Vertex Cycles • Cycle – a path that starts and ends on the same vertex with no repeated edges • Vertex Cycle – a cycle that contains every vertex • Unsolved problem: Identifying exactly when a vertex cycle exists.

17. 4.4 Vertex Cycles • Dirac’s Theorem – if the following are all true for a graph then a vertex cycle must exist: • Graph is connected • Graph has 3 or more vertices • Each vertex is adjacent to at least half of the vertices • Note: Dirac’s conditions may not be satisfied and yet a vertex cycle still exists.

18. 4.4 Vertex Cycles • Complete Graph – every vertex is connected to every other vertex • Definition: • Theorem: A complete graph with n vertices has (n-1)! vertex cycles • Note: This becomes if reverse cycles are not included.

19. 4.5 Traveling Salesman Problem • Not the “how to get rid of a traveling salesman” problem • Problem: How to schedule visits to cities in the cheapest way possible – using graph theory • Weighted Graph – a weight (number) is assigned to each edge • Graph theory problem: Finding the vertex cycle with the smallest total weight

20. 4.5 Traveling Salesman Problem • Solution – Brute Force Method: • List all vertex cycles (reverse cycles not included) • Find the total weight of each cycle • Pick the cycle with the smallest weight • Problem with this approach: cycles must be checked – gets large in a hurry

21. 4.5 Traveling Salesman Problem • Lower Bound – Add the weights of the n shortest edges where n = # of vertices • Applications of traveling salesman problem: • Deliveries (e.g. Federal Express) • Collections(e.g. Trash Pickup) • Manufacturing(drilling holes in circuit boards – need to minimize the movement of the drill)

22. 4.6 Approximations to Traveling Salesman Problem • Nearest Neighbor Approximation: • Select the starting vertex. • Select the unused connecting edge with the smallest weight. • Repeat step 2 until all vertices are reached. • Select the edge that goes back to the starting point to complete the cycle

23. 4.6 Approximations to Traveling Salesman Problem • Cheapest Link Approximation: • Select the edge with the smallest weight. • Select the unused edge with the smallest weight unless it completes a cycle or one vertex becomes connected to 3 edges. Break ties arbitrarily. • Repeat step 2 until all vertices have been included in a cycle.

24. 4.6 Approximations to Traveling Salesman Problem - Comparison • Nearest Neighbor Approximation: • Starting vertex must be specified. • Order selected is always sequential – starting vertex is always the end vertex of the previous edge. • Cheapest Link Approximation: • Order selected is not necessarily sequential. • Weights are always increasing.

25. 4.7 Trees • Tree – a graph that is connected and has no cycles. Examples: family tree, organizational chart • Properties of trees: • For any 2 vertices there is a unique path between them • A tree with n vertices has n-1 edges • A tree has the least number of edges possible for a connected graph

26. 4.7 Trees • Determining whether a graph is a tree -A graph with n vertices and n-1 edges must be a tree • Spanning Tree of a Graph – a tree that is a subgraph that contains all the vertices of the graph • Minimal Spanning Tree – the spanning tree of a graph with the smallest weight

27. 4.7 Trees • Kruskal’s method for finding the minimal spanning tree: • Find the edge with the smallest weight • Find the next smallest unused edge that does not form a cycle • Repeat step 2 until all vertices are included. Remember - do not complete a cycle!

28. 4.8 Instant Insanity • Instant Insanity - uses 4 blocks with colors on the sides (red, white, blue, green). • Goal – to stack the cubes so that all 4 colors occur exactly once on each side. • There are 331,776 possibilities with trial and error. • The puzzle can be solved using graph theory.

29. 4.8 Instant Insanity • Cubes must be represented in 2 dimensions • Convert each cube to a graph representation by connecting colors of opposite sides R W R B G W W G B G

30. 4.8 Instant Insanity • Consolidate to one graph, labeling paths by cube number 2 R W 3 R W R W R W R W 1 1 2 2 3 4 1 3 B G B G B G B G 4 B G 4

31. 4.8 Instant Insanity • Solution to consolidated graph – pick 2 subgraphs such that • Each vertex has degree 2 • Each cube is represented exactly once in each subgraph (1, 2, 3, and 4) • 2 subgraphs have no edge in common (same number/edge in both subgraphs) • Note: forms of solutions are given in book

32. 4.8 Instant Insanity • Solution subgraphs for the example: 2 R W 3 R W R W 1 1 1 2 2 4 4 3 3 1 2 2 3 4 1 3 4 B B G G B G 4