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Section 14.1

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 bridges exactly once? Graph theory provides a way to mathematically answer that question. Konigsberg.

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Section 14.1

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  1. Section 14.1 Intro to Graph Theory

  2. Beginnings of Graph Theory • Euler’s Konigsberg Bridge Problem (18th 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 ofdegree 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. 14.2.

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

  11. The edge DE is a bridge E A D F B C 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?

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