Euler and Hamilton circuits. Euler paths and circuits

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1 uler and Hamilton circuits uler paths and circuits o The Seven ridges of Konigsberg In the early 1700 s, Konigsberg was the capital of ast Prussia. Konigsberg was later renamed Kaliningrad and is now in Russia. The Pregel river runs through the city and branches into two rivers with an island (labeled ) in the middle as shown: Problem: Is it possible for a citizen of Koningsberg to take a stroll through the city, crossing each bridge exactly once, and beginning and ended at the same place? Simplify the problem by drawing a graph whose vertices represent the four land masses (,,, ) and whose edges represent the bridges connecting them. Then try to find a circuit that uses each edge exactly once. o n uler path is o n uler circuit is

2 2 xample 1. onsider the following graph: (a) Is an uler circuit? (b) an you find an uler circuit? uler s Theorem: In any connected graph: If the graph has an uler circuit, If each vertex of the graph has even degree, o xample 2. What does uler s Theorem tell us for the Seven ridges problem? o xample 3. ecide which of the following graphs have an uler circuit: L K G M I J H H J (a) G (b)

3 3 o xample 4. an you draw the following graph without lifting your pencil and without redrawing any of its edges? leury s lgorithm o cut edge in a graph is xample 5. Identify all cut edges in the graph in the following graph: G H

4 4 o leury s lgorithm for finding an uler circuit when all vertices have even degree: 1. Start at any vertex and choose any edge from this vertex. Go along your chosen edge to its other vertex and remove this edge from the graph. 2. You are now at a vertex of the revised graph. If the only edge from this vertex is a cut edge, go along this edge to its other vertex and remove this edge. Otherwise, choose any edge from this vertex that is not a cut edge, go along this edge to its other vertex and remove this edge. 3. Repeat step 2 until all edges have been used and you have returned to the vertex from which you started. xample 6. Use leury s lgorithm to find and uler circuit in the following graph: H G Hamilton ircuits o Hamilton circuit in a graph is

5 5 xample 1. Which of the circuits (a) (c) below are Hamilton ircuits? Which are uler circuits? (a) (b) (c) very graph with has a Hamilton circuit o Hamilton circuits that differ only in their are thought of as xample 2. raw a complete graph and count the number of Hamilton circuits. Number of Hamilton circuits a) 3 vertices b) 4 vertices c) 5 vertices

6 6 o The number of Hamilton circuits in a complete graph with n vertices is o In a weighted graph, a minimum Hamilton circuit is o rute orce lgorithm for finding a minimum Hamilton circuit 1. hoose a starting point. 2. List all Hamilton circuits with that starting point. 3. ind the weight of each circuit. 4. hoose a Hamilton circuit with the least total weight. xample 3. Use the rute orce lgorithm to ind a minimum Hamilton circuit for the following complete weighted graph: o The rute orce lgorithm is inefficient because it requires us to calculate the total weight for each of the (n 1)! Hamilton circuits. o No efficient algorithm is known, but an algorithm that gives reasonably good solutions most of the time is the

7 7 o Nearest Neighbor lgorithm for finding a minimum Hamilton circuit 1. hoose a starting point, call it. 2. rom the edges connecting to vertices not yet visited, choose the one with smallest weight and proceed along this edge to its other vertex. 3. Repeat step 2 until all vertices have been visited. 4. Return to the starting vertex. o The Nearest Neighborhood lgorithm if an example of an xample 4. courier based at head office and must deliver documents to four other offices,,, and. The estimated travel time (in minutes) between each office is shown on the graph below. The courier wishes to visit the locations in an order that takes the least amount of time. Use the Nearest Neighbor lgorithm to find an approximate solution to this problem, and calculate the total time for the given route

8 8 xample 5. You might be able to improve your result in xample 4 by starting at a different vertex: Starting vertex ircuit using Nearest Neighbor lgorithm Total weight The worst solution given by the Nearest Neighbor lgorithm begins at vertex