Connected Graphs and Connectivity prepared and instructed by
Connected Graphs and Connectivity prepared and instructed by Shmuel Wimer Eng. Faculty, Bar-Ilan University May 2014 Connectivity 1
The Friendship Theorem May 2014 Connectivity 2
impossible May 2014 Connectivity 3
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Euler Tours May 2014 Connectivity 8
Lemma. Every maximal trail in an even graph is closed. May 2014 Connectivity 9
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G May 2014 Connectivity 12
Iteration: Traverse from the current vertex any non cutedge, unless there is no other alternative. May 2014 Connectivity 13
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? May 2014 Connectivity 15
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Application: Layout of CMOS Gates May 2014 polysilicon diffusion metal contact Connectivity 17
The Chinese Postman Problem Find a closed walk of minimum weight that traverses all the edges. Edges are duplicated to produce an even graph. The problem is therefore to minimize the total weight of edge duplicates producing an even graph. May 2014 Connectivity 18
Edges need not be multiplied more than once. (why? ) 4 1 3 3 4 1 7 2 2 1 7 4 1 1 3 3 1 4 Duplicated edge connecting odd and even vertices switch their evenness. May 2014 Connectivity 19
Edge addition must proceed until an odd vertex is met. (Edmonds and Johnson 1973). If there were only two odd vertices, a shortest path connecting those solves the problem. May 2014 Connectivity 20
Connection in Digraphs May 2014 Connectivity 21
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Algorithm. (Directed Eulerian Tour) May 2014 Connectivity 24
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Application. Testing the position of a rotating drum. 0 0 1 1 May 2014 1 1 0 1 Connectivity 1 30
0 1 0 1 1 The problem can be solved using Eulerian digraph. May 2014 Connectivity 31
001 1 0 000 1 010 0 0 100 011 1 0 1 1 101 0 0 111 1 110 011 100 0 110 0 101 Connectivity 1 1 0 1 0 1 1 tail vertex 0 1 111 May 2014 1 0 0 0 1 0 0 head vertex 32
tail vertex head vertex May 2014 Connectivity 33
Application. Street-Sweeping Problem. May 2014 Connectivity 34
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Transportation problem was introduced by Kantorovich (1939), solved by Hitchcock (1941) and many others. ■ May 2014 Connectivity 37
Cuts and Connectivity It is desired to preserve network service when some nodes or connections break. For expensive connections it is desired to maintain connectivity preservation with as few edges as possible. Graphs and digraphs are assumed loopless. May 2014 Connectivity 38
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Edge Connectivity May 2014 Connectivity 43
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A communication network is fault-tolerant if there alternative paths between vertices. The more vertex disjoint paths (except ends) the better. May 2014 Connectivity 48
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Proof. Suppose that any two vertices are connected with a pair of internally-disjoint paths. Deletion of one vertex cannot disconnect these vertices, and at least two vertex deletion is required, hence G is 2 connected. May 2014 Connectivity 50
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