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Section 14.1 Intro to Graph Theory

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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

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

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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.

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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

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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?

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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?

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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

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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.

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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

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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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