1. Graph theory at work

2.  Examine the procedure utilized by snow plows in Iowa City  Systemize and minimize routes  Review mathematical concepts involved  Look into how math concepts apply to this problem  Model  Apply to example section  Conclusions/Recommendations

3.  Winter of 2007-2008  Unplowed areas  Results  Uneven roads  Unable to plow  Cracks and potholes

4.  Environmental  Reduce Gas Consumption  Greenhouse gas emissions  Save money  Public  Complaints  City website  Safety

5.  System  Current process  Downtown/Bus routes  Steep slopes  Flat secondary roads  Easy to teach  Little confusion

6.  Seven Bridges of Könisburg  Euler

7.  Traverse each edge exactly once  Circuits exist if all vertices of even degree  Digraph: indegree equals outdegree for all vertices  Use here  If one exists, will be optimal route  More than one truck

8.  Multigraph  Vertices - intersections of roads  Edges – bidirectional streets  Directed arcs – one-way streets  For snow plows  Must traverse each lane of each road at least once  Digraph  Vertices – intersections of roads  Arcs – directed lanes

9.  Kwan Mei-Ko 1960’s  Goal: traverse every street in least distance  More general than bridge problem  If contains eulerian circuit, this is the shortest route  If not, solution can be found

10.  If not using city’s current priorities  Weights represent distance  Want Mininimum  Find degrees of all vertices in graph  Must be even number of vertices of odd degree  Handshake lemma  Find shortest weighted paths between these vertices  Draw duplicate edges along path  Will then have all even degrees  Create Eulerian Circuit

11.  If I choose to comply with current process  Assign weights to streets  Weights represent grade of street  Find maximal weighted paths first  Represent steep slopes  Follow by lower weighted paths  Flatter streets

12.  Square matrix  Each row and column represents a vertex  ‘1’ in X ij if arc (edge) exists from i to j  ‘0’ otherwise  Will be used to find degree of vertex in multigraph  Sum of ones in vertex row/column (digraph)

13.  Traverse bus routes  Simple because already circuits  Divide city into sections (10)  Within each section, split roads into phases  Maintain city’s current priorities  Each phase  Create adjacency matrix  For vertices of odd degree, create connected graph with weights of shortest distance between  Find perfect matching  New edges along path  Find Eulerian circuit  Repeat for steep roads, flat roads

14. 1 2 4 3 5 6 7 8 Vertices 2,3,4,5,6,7 have odd degree

15. 2 4 3 5 6 7 Assign weights (shortest distance) Find minimal matching 1 1 1 1 4 3

16. 2 4 3 5 6 7 1 1 1 1 4 3 Duplicate red edges

17. 2 4 3 5 6 7 1 8

18. 2 4 3 5 6 7 1 8

19. 2 4 3 5 6 7 1 8

20. 2 4 3 5 6 7 1 8

21. 2 4 3 5 6 7 1 8

22. 2 4 3 5 6 7 1 8

23. 2 4 3 5 6 7 1 8

24. 2 4 3 5 6 7 1 8

25. 2 4 3 5 6 7 1 8

26. 2 4 3 5 6 7 1 8

27. 2 4 3 5 6 7 1 8

28. 2 4 3 5 6 7 1 8

29. 2 4 3 5 6 7 1 8

30.  Divide the city into sections  Determine which streets fall into which phase  Determine distances between vertices  Create computer program  Takes in vertices/edges  Forms adjacency matrices  Finds degrees  Forms weighted matrix for vertices of odd degree  Minimizes matching  Duplicates these edges  Results in minimal distance path for each phase