Simple Graphs: if they remain connected k-Connectivity whenever - PowerPoint PPT Presentation

Edge Connectedness Mathematics for Computer Science MIT 6.042J/18.062J Def: vertices v, w are k-edge connected Simple Graphs: if they remain connected k-Connectivity whenever fewer than k edges are deleted. Albert R Meyer, April 5, 2013 Albert R Meyer, April 5, 2013 k-connect.1 k-connect.2 k-edge Connectedness k-edge Connectedness no path delete 1-edge connected 1-edge connected Albert R Meyer, April 5, 2013 Albert R Meyer, April 5, 2013 k-connect.3 k-connect.4 1

Edge Connectedness Edge Connectedness no path 2-edge connected 2-edge connected Albert R Meyer, April 5, 2013 Albert R Meyer, April 5, 2013 k-connect.5 k-connect.6 Edge Connectedness Edge Connectedness no path 3-edge connected 3-edge connected Albert R Meyer, April 5, 2013 Albert R Meyer, April 5, 2013 k-connect.7 k-connect.8 2

Edge Connectedness k-edge Connectedness Connectivity measures fault Def: A graph is tolerance of a network: k-edge connected how many connections can iff every two vertices fail without cutting off are k-edge connected. communication? Albert R Meyer, April 5, 2013 Albert R Meyer, April 5, 2013 k-connect.9 k-connect.10 k-edge Connectedness k-edge Connectedness this whole graph is this whole graph is delete 1-edge connected 2-edge connected Albert R Meyer, April 5, 2013 Albert R Meyer, April 5, 2013 k-connect.11 k-connect.12 3

k-vertex Connectedness k-vertex Connectedness k-vertex connected k-vertex IMPLIES k-edge connected connectedness not conversely: defined similarly 2-edge connected 1-vertex connected Albert R Meyer, April 5, 2013 Albert R Meyer, April 5, 2013 k-connect.13 k-connect.14 k-vertex Connectedness k-vertex Connectedness The n-dimensional K n is the complete hypercube H n graph on n vertices. V(H n ) ::= {0,1} n K n is (n-1)-vertex an edge IFF u,v u − v connected. differ in 1 place Albert R Meyer, April 5, 2013 Albert R Meyer, April 5, 2013 k-connect.15 k-connect.16 4

Menger’s Theorem k-vertex Connectedness k-connected vertices H n is n-vertex will be connected by k connected. non-overlapping paths (class problem) Albert R Meyer, April 5, 2013 Albert R Meyer, April 5, 2013 k-connect.17 k-connect.18 5

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