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Section 14.1 Intro to Graph Theory. Beginnings of Graph Theory Euler’s Konigsberg Bridge Problem (18 th c.)  Can one walk through town and cross all.

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Presentation on theme: "Section 14.1 Intro to Graph Theory. Beginnings of Graph Theory Euler’s Konigsberg Bridge Problem (18 th c.)  Can one walk through town and cross all."— Presentation transcript:

1 Section 14.1 Intro to Graph Theory

2 Beginnings of Graph Theory Euler’s Konigsberg Bridge Problem (18 th c.)  Can one walk through town and cross all bridges exactly once?  Graph theory provides a way to mathematically answer that question

3 Konigsberg Two islands connected to land and each other by 7 bridges:

4 Representing the problem The Konigsberg problem can be represented by a graph A DOT is a VERTEX. In this problem, a vertex represents a land mass. A line is an EDGE. Land masses are connected by EDGES if they are linked (by a bridge, in this problem) Two vertices are ADJACENT if they share an edge. Land masses are adjacent if they are connected by a bridge.

5 Terminology VERTEX  There are 4 vertices EDGE  Vertex D has 3 edges DEGREE  Vertex D is of degree 3  Vertex A is of degree 5 ADJACENT  Vertex A is adjacent to vertex D  Vertex C is not adjacent to vertex B

6 More Terminology Odd vs. Even  A vertex is ODD if its degree is an odd number  Likewise, a vertex is EVEN if its degree is an even number  Is A odd or even?  Is C odd or even?

7 Back to the question… Can you walk on each bridge exactly once?  Try using the graph and a pencil: Trace a route without picking up your pencil.  What did you find?

8 Solving the bridge problem What do you notice about the degree of all the vertices? Are the vertices odd or even? We will solve this problem in Sec

9 Moving on a graph PATH: a sequence of adjacent vertices and the edges connecting them.  In the graph above, an example of a path is C, D, B. CIRCUIT: Path that begins and ends at the same vertex.  In the graph above, an example of a circuit is A, D, B, A. In Konigsberg, the problem was to find a CIRCUIT that uses every edge.

10 Connected vs Disconnected A graph is CONNECTED if there is a path between any two vertices of the graph. This Graph is DISCONNECTED B C D A F E This Graph is CONNECTED B C D A F E

11 Making a Graph Disconnected A BRIDGE is an edge that if removed from a connected graph, it would disconnect the graph. The edge DE is a bridge  There are two other bridges in this graph. Can you find them? The edge DE is a bridge B C D A F E


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