Graph Theory

1. Graph Theory Thinking Mathematically, 10.7 & 15.1–15.3 Excursions in Mathematics, 5.1–5.7

2. Origins of graph theory Thinking Mathematically, Section 10.7

3. ORIGINS OF GRAPH THEORY • Seven Bridges • Salesman’s Bridge • Green Bridge • Slaughter Bridge • Blacksmith’s Bridge • Timber Bridge • High Bridge • Honey Bridge A K L • Four City Areas • Alstadt & Löbnicht • Kneiphof Island • Vorstadt • Lomse Island V Pregel River Königsberg, Germany

4. ORIGINS OF GRAPH THEORY • Five Bridges Today • Estacada Bridge • Timber (October) Bridge • High Bridge • Honey Bridge • NEW: Emperor Bridge Kaliningrad, Russia

5. ORIGINS OF GRAPH THEORY • Origins of Graph Theory • Königsberg, Germany in early 1700s had seven bridges • Residents of city took walks on Sunday, developed a game • “Can a person cross all bridges only once and return to start?” • 1735 Leonhard Euler was given the problem and determined impossible • Leonhard Euler • Swiss mathematician • St. Petersburg Academy of Sciences • Started “geometrissitus” which became graph theory

6. In some textbooks, the number of edges is called its degree. DEFINITIONS vertex – point (NOTE: plural is vertices) edge – line segment or curve that starts at one vertex and ends at another vertex odd vertex – vertex with an odd number of attached edges even vertex – vertex with an even number of attached edges traversable graph – a graph for which all edges can be traced once and only once

7. EXAMPLE Consider the graph. Is it traversable? That is, can each line of the graph be traced once and only once without lifting your pencil? Yes, the graph is traversable!

8. Rules Euler’s Rules of Traversability 1. A graph with all even vertices is traversable. One can start at any vertex and end at same vertex. 2. A graph with two odd vertices is traversable. One must start at one odd vertex and end at the other odd vertex. 3. A graph with more than two odd vertices is NOT traversable.

9. Back to EXAMPLE Odd (3) Odd (3) There are two, and only two, odd vertices so it was traversable. Even (2) Consider the graph. Is it traversable? That is, can each line of the graph be traced once and only once without lifting your pencil? Yes, the graph is traversable! Had to start at an odd vertex!

10. Recall Euler’s Rules of Traversability: 1. A graph with all even vertices is traversable. One can start at any vertex and end at same vertex. EXAMPLE Even (2) Even (2) Even (2) Even (4) Even (2) Consider the graph. Is it traversable? Yes, the graph is traversable!

11. Let’s model the Seven Bridges of Königsberg as a graph. Is it traversable? Let’s find out…

12. Recall Euler’s Rules of Traversability: 3. A graph with more than two odd vertices is NOT traversable. EXAMPLE Odd (3) The Bridges of Königsberg problem was not a traversable graph. Odd (5) Odd (3) Odd (3) Consider the graph. Is it traversable? No, the graph is not traversable!

13. EXAMPLE Consider the graph. Is it traversable? If so, find the path. Yes, the graph is traversable!

14. EXAMPLE Consider the graph. Is it traversable? If so, find the path. No, the graph is not traversable!

15. EXAMPLE Consider the graph. Is it traversable? If so, find the path. No, the graph is not traversable!

16. Graphs, paths, & circuits Thinking Mathematically, Section 15.1

17. DEFINITIONS point must be marked with a dot; denoted with letter denoted with two vertex letters denoted with repeated vertex letter vertex – point (NOTE: plural is vertices) edge – line segment or curve that starts at one vertex and ends at another vertex loop – curve that starts and ends at the same vertex

18. Not every point where two edges cross is a vertex! If the point is not marked with a dot, it is not a vertex (think of it as one line is above the other). EXAMPLE 4 How many vertices? 4 How many edges? edge AD edge DB edge BC edge CA This is NOT a vertex! 1 How many loops? Count lines going out from C…does not matter if two connect. loop CC Consider the graph below. Identify the vertices, edges, and loops. What is C’s degree? 4

19. DEFINITION equivalent graphs – two graphs with the same number of vertices connected to each other in the same way

20. The placement of the vertices and the shapes of the edges does NOT matter…for graphs to be equivalent, only how the vertices connect to one another matters! EXAMPLE 2 1 4 3 Which are equivalent graphs?

21. EXAMPLE Consider the Seven Bridges of Königsberg problem, modeled below. Draw an equivalent graph.

22. Now we will model relationships using graphs…since several graphs may be equivalent, there will often be more than one correct answer when modeling! EXAMPLE Consider the map of the Rocky Mountain states. Model the states that share a common border.

23. Recall Euler’s Rules of Traversability: 2. A graph with two odd vertices is traversable. One must start at one odd vertex and end at the other odd vertex. EXAMPLE Yes, it is traversable. Even (2) Odd(3) Even (4) Odd(3) Even (2) Can you cross the border of each state once and only once (that is, is the graph traversable)?

24. EXAMPLE Even (2) Odd(3) Even (4) Odd(3) Even (2) Find the path that traverses the graph. Find the path on the map that crosses each border exactly one time. A graph that goes to each state (thus each dot) only once is a different kind of graph…that’s for later.

25. EXAMPLE Consider the map of the New England states. Model the states that share a common border.

26. Recall Euler’s Rules of Traversability: 2. A graph with two odd vertices is traversable. One must start at one odd vertex and end at the other odd vertex. EXAMPLE Yes, it is traversable. Odd (1) Even (2) Odd (3) Even (4) Even (2) Even (2) Can you cross the border of each state once and only once (that is, is the graph traversable)?

27. EXAMPLE Odd (1) Even (2) Odd (3) Even (4) Even (2) Even (2) Find the path that traverses the graph. Find the path on the map that crosses each border exactly one time.

28. Now let’s focus just on the modeling itself… EXAMPLE Consider the floor plan of a four-room house. (Consider all the outdoors as one location.) Model the ways to walk from one area to another.

29. EXAMPLE Consider the floor plan of a four-room house. (Consider all the outdoors as one location.) Model the ways to walk from one area to another.

30. EXAMPLE Consider the bridges of Madison County. Model the ways to walk from one area to another.

31. EXAMPLE Consider this neighborhood. A mail carrier delivers the mail by walking to each house. If houses are on both sides of the street, he walks it twice. Model this. Houses are purple squares. Roads are pink lines. Mark intersections & turns. Make connections.

32. Vertices that are near each other are not necessarily adjacent …there must be an edge between them! DEFINITIONS D is adjacent to C C is adjacent to A & D B is adjacent to A & E A is adjacent to B, C, E, & F E is adjacent to A & B F is adjacent to A adjacent vertices – vertices that are connected directly and thus share at least one edge F is NOT adjacent to E

33. NOTE: A path can use a vertex more than once but can use each edge only once. However, the entire graph does not have to be touched to have a path. DEFINITIONS path: A,B,F,G,H,M path: A,B,F,K,L,G,F,E not a path: A,B,F,G,H,M,H,D path – a sequence of adjacent vertices and the edges connecting them, denoted by list of vertices in order

34. NOTE: Just like a path, a circuitcan use a vertex more than once but can use each edge only once. However, the entire graph does not have to be touched to have a circuit. DEFINITIONS circuit: A,B,F,G,L,K,J,E,A NOTE: Every circuit is a path. Every path is not a circuit! circuit – a path that begins and ends at the same vertex

35. DEFINITIONS disconnected graph There is no path from the light blue dot to the dark blue dot. connected graph – a graph in which there is at least one path connecting any two vertices disconnected graph – a graph in which there is no path connecting any two vertices

36. DEFINITIONS edge BC is a bridge edge AD is not a bridge bridge – an edge that, if removed, would make a connected graph into a disconnected graph

37. EXAMPLE edge AB is a bridge edge BG is a bridge Find the bridge(s) in the graph.

38. HOMEWORK From the Cow book 10.7 pg 549 # 1 – 6 all 15.1 pg 786 # 1 – 47 odd