[PDF] [PDF] 83 Hamiltonian Paths and Circuits

[PDF] 83 Hamiltonian Paths and Circuits

Then G has a Hamiltonian circuit if m ≥ ½(n2 – 3n + 6) where n is the number of vertices Page 5 Hamilton Paths and Circuits A is a continuous path that passes

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An Euler path exists exist i there are no or zero vertices of odd degree Proof ): An Euler circuit exists As the respective path is traversed, each time we visit a 

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This is an example of a Graph Theory problem that needs solving What you need is called a Hamiltonian circuit : it's a path around the suburb that stops at each 

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Euler paths and circuits contained every edge only once Now we look at paths every vertex exactly once is called a Hamilton path, and a simple circuit in a 

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Definition A Hamiltonian cycle (or circuit) is a closed path that visits each vertex once Definition A graph that has a Hamiltonian cycle is called Hamiltonian

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Hamilton Path is a path that contains each vertex of a graph exactly once Hamilton Circuit Some books call these Hamiltonian Paths and Hamiltonian Circuits

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A path or circuit P in a (directed) graph G is called Hamiltonian provided P is simple and contains all the vertices of G An n-tournament is an oriented complete 

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Properties of Euler's Graph 1 Theorem 1: A connected multigraph with at least two vertices has an Euler Circuit if and only if each of its vertices has even 

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Hamiltonian Circuit: A Hamiltonian circuit in a graph is a closed path that visits every vertex in the graph exactly once (Such a closed loop must be a cycle ) A 

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8.3Hamiltonian Paths and Circuits

8.3 -Hamiltonian Paths and Circuits•A Hamiltonian pathis a path that contains each vertex exactly once•A Hamiltonian circuitis a Hamiltonian path that is also a circuit

8.3 -Hamiltonian Paths and Circuits•Theorem 8.3.1•Let G be a connected graph with n vertices (n ÎZ, n >2), with no loops or multiple edges. G has a Hamiltonian circuit if for any 2 vertices u and v of G that are not adjacent, the degree of u plus the degree of v is ³n.•Corollary 8.3.1•G has a Hamiltonian circuit if each vertex has degree ³(n/2).

8.3 -Hamiltonian Paths and Circuits•Theorem 8.3.2•Let the number of edges of G be m. Then G has a Hamiltonian circuit if m ³½(n2-3n + 6) where n is the number of vertices.

Hamilton Paths and Circuits A ______________ is a continuous path that passes through every _________ once and only once.A _______________ is a Hamilton path that begins and ends at the same vertex. (the starting/end vertex will be the onlyvertex touched twicevertexHow is a Hamilton Path different from a Euler path or Circuit?Hamilton PathHamilton Circuit

6Hamilton circuits and paths•A Hamilton circuit(path) is a simple circuit (path) that contains all vertices and passes through each vertex of the graph exactly once.•How can we tell if a graph has a Hamilton circuit or path?•Not easily, i.e., in general, in not less than exponential time in the number of vertices•There are some tests that can exclude graphs (e.g., no vertex of degree one for circuits).

7Ham. Circuits and Paths

8Number of Hamilton paths•Consider the complete graph Knwith n> 2. How many Hamilton circuits are there?•Select any vertex as the start vertex (because all vertices will belong to the circuit the choice doesn't matter). From this vertex we can choose n-1 successor vertices, from each of them n-2 vertices, and so on, for a total of (n-1)! circuits. Because direction doesn't matter, the distinct circuits are (n-1)!/2.

9Weighted graphs•A weightedgraph is a graph where numbers (weightsor costs) have been attached to each edge.•Example: In a graph for flight connections, weights could represent time needed for each flight, distance traveled, or cost of traveling on that flight. In a computer network, costs could represent the delay for a message to travel through a link.

10Path length•In a graph without weights, we define the length of a path as the number of edges in it.•In a weighted graph, the path length is a function of the weights of the edges in the path, usually the sum of those weights.

11The Traveling Salesman problem•A traveling salesman needs to visit ncities, going to each city exactly once, and return to his starting city. Assume every city is connected to every other city, but the cost of traveling differs for each city pair. What is the sequence of cities that minimizes the overall cost?•Equivalent formulation: Find the Hamilton circuit in Knwith shortest length.

12Solving the TSP•No algorithm is known that will guarantee finding the best solution except by examining all possible circuits (or equivalent approaches).•For n = 25, (n-1)! / 2 = 24! / 2 ≈3 ×1023. If we take one nanosecond to calculate the cost of each circuit, we still need 3 ×1014seconds, or about 10 million years.•The problem is provably NP-complete.

13TSP Problems•Planning routes•980 cities in Luxembourg•24,978 cities in Sweden (solved May 2004)•1,904,711 cities in the world•Drilling printed circuit boards•cost of moving the drill and possibly changing drills because of different hole sizes•Genome sequencing•edges represent likelihood that a fragment of DNA follows another fragment

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