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9.1 A Walk Through Konigsberg
Is it possible for a pedestrian to walk across each of the 7 bridges exactly once?
Leonard Euler (rhymes with
Graph – diagram consisting of vertices, and edges
Vertices (vertex) –
Edges –
Loops –
Two graphs are identical if the number of vertices and the edges connecting them are
identical.
Ex a Are the graphs below identical?
Family tree graph
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9.2 Graphs and Euler Trails (thanks to Heidi Meyer for her graph library)
A graph is traversable if all edges can be traced continuously (without lifting your pen) and no
edge is traced more than once.
Ex a Which of the following are traversable?
degree of a vertex – number of edges that connect to that vertex
Exb Find the degree of each of the vertices in the examples above. Do you see a
connection between the degree of the vertices and traversability?
adjacent vertices – joined by an edge
adjacent edges – share a vertex
identical graphs – have the same vertices, and the same # of edges connecting each pair of
vertices
trail – a sequence of edges connecting adjacent vertices
circuit – a trail that begins and ends at the same vertex
Euler trail – a trail that travels through every edge exactly once (may or may not start/stop in
the same place
Euler circuit – an Euler trail that is a circuit (starts/stops in the same place)
Ex b Find the degree of each of the vertices of the graphs below.
Which of the following have Euler trails or Euler circuits?
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Eulerization – to create an Euler trail or Euler circuit:
1. Locate the odd degree vertices.
2. Connect odd degree vertices until 2 or 0 odd vertices remain
 If direct connections are allowed, use them.
 If not, use the smallest number of duplicate edges
(You are not required to use Fleury’s Algorithm or the Eulerization Algorithm)
Ex c Create Euler circuits for the following graphs:
Applications of Euler Circuits
Ex d From the BART Map (https://www.bart.gov/stations), create a graph (use map as if
there were no time limits). Is there an Euler trail or circuit?
Homework Assignment # 11 – Due Nov. 10, 2015
From your textbook, complete the following homework problems:
Section 9.1 # 7, 9, 10, 11, 12, 16
Section 9.2 # 2, 3, 5, 6, 14, 15, 17, 18, 23, 26, 29