ch05 - Excursions in Modern Mathematics Sixth Edition Peter...

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1 Excursions in Modern Mathematics Sixth Edition Peter Tannenbaum
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2 Chapter 5 Euler Circuits The Circuit Comes to Town
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3 Euler Circuits Outline/learning Objectives To identify and model Euler circuit and Euler path problems. To understand the meaning of basic graph terminology. To classify which graphs have Euler circuits or paths using Euler’s circuit theorems.
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4 Euler Circuits Outline/learning Objectives (cont.) To implement Fleury’s algorithm to find an Euler circuit or path when it exists. To eulerize or semi-eulerize graphs when necessary. To recognize an optimal eulerization (semi- eulerization) of a graph.
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5 Euler Circuits 5.1 Euler Circuit Problems
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6 Euler Circuits Existence question Existence question Is an actual route possible? Optimization question Optimization question Of all the possible routes, which one is the optimal route ? What is a routing problem?
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7 Euler Circuits We will answer both the existence and optimization questions for a special class of routing problems known as Euler circuit problems . The common thread is what we call the exhaustion requirement.
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8 Euler Circuits The name of the game is to trace each drawing without lifting the pencil or retracing any of the lines . These kinds of tracings are called unicursal tracings .
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9 Euler Circuits When we end in the same place we started, we call it a closed unicursal tracing; when we start and end in different places, we call it an open unicursal tracing. .
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10 Euler Circuits 5.2 Graphs
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11 Euler Circuits Vertices Vertices - dots - dots Edges Edges - lines - lines The edges do not have to be straight lines. But they The edges do not have to be straight lines. But they have to connect two vertices. have to connect two vertices. Loop Loop - an edge connecting a vertex back with itself - an edge connecting a vertex back with itself A graph is a picture consisting of:
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12 Euler Circuits This graph has six vertices A, B, C, D, E, and F and eight edges. The edges can be described by giving the two vertices that are connected by the edge. Thus the edges are AB, AD, BB, BC, BE, CD, CD, and DE
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13 Euler Circuits First, note that the point where edges BE and AD cross is not a vertex– it is just the crossing point of two edges. Second, that vertex F is not connected to any other vertex. Such a vertex is called an isolated vertex.
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14 Euler Circuits Third, note that this graph has a loop, namely the edge BB. Finally, note that it is permissible to have two edges connecting the same two vertices, as in the case with C and D. When a graph has more than one edge connecting the same pair of vertices, it is said to have multiple edges.
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15 Euler Circuits This graph is considered a single graph, even though it consists of two separate, disconnected pieces. Such graphs are called disconnected graph, and the individual pieces are called the components of the graph. .
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16 Euler Circuits A Graph with No Edges? Yes, its possible. Without edges, every vertex of the graph is an isolated vertex .
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This note was uploaded on 11/11/2011 for the course MATH 110 taught by Professor Staff during the Winter '08 term at BYU.

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ch05 - Excursions in Modern Mathematics Sixth Edition Peter...

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