Trees v w a AlbertRMeyer, April8,2013 AlbertRMeyer, - PowerPoint PPT Presentation

Feb 21, 2024 •146 likes •215 views

Cycles in simple graphs Mathematics for Computer Science MIT 6.042J/18.062J Acycleisaclosedwalk oflength > 2thatdoesnt crossitself: Trees v w a AlbertRMeyer, April8,2013 AlbertRMeyer, April8,2013 tree-def.1

Cycles in simple graphs Mathematics for Computer Science MIT 6.042J/18.062J A cycle is a closed walk of length > 2 that doesn’t cross itself: Trees v w a Albert R Meyer, April 8, 2013 Albert R Meyer, April 8, 2013 tree-def.1 tree-def.2 Cycles in simple graphs Trees length > 2 implies that A tree is a connected graph going back & forth over with no cycles. an edge is not a cycle w v Albert R Meyer, April 8, 2013 Albert R Meyer, April 8, 2013 tree-def.3 tree-def.4 1
Come up all the time More Trees • family trees • search trees • game trees • parse trees • spanning trees Albert R Meyer, April 8, 2013 Albert R Meyer, April 8, 2013 tree-def.5 tree-def.6 Lots of kinds Focus on “pure” trees • rooted unordered, • ordered unrooted, • binary undirected • complete • directed Albert R Meyer, April 8, 2013 Albert R Meyer, April 8, 2013 tree-def.7 tree-def.9 2
Cut Edges Cut Edges An edge is a cut edge if removing it from the graph disconnects two vertices. Albert R Meyer, April 8, 2013 Albert R Meyer, April 8, 2013 tree-def.10 tree-def.11 Cut Edges Cut Edges e deleting e gives e is a cut edge two components Albert R Meyer, April 8, 2013 Albert R Meyer, April 8, 2013 tree-def.12 tree-def.13 3
Cut Edges Cut Edges f still connected with f is not a cut edge edge f deleted Albert R Meyer, April 8, 2013 Albert R Meyer, April 8, 2013 tree-def.14 tree-def.15 Cut Edges Cut Edges and Cycles So a connected graph is Lemma: An edge is a 2-edge connected iff not a cut edge iff it is on a cycle. it has no cut edge. Albert R Meyer, April 8, 2013 Albert R Meyer, April 8, 2013 tree-def.16 tree-def.17 4
alternative tree definition alternative tree definition A tree is a connected graph A tree is a connected graph with every edge a cut edge. that is edge-minimal. equivalently equivalently Albert R Meyer, April 8, 2013 Albert R Meyer, April 8, 2013 tree-def.18 tree-def.19 more equivalent definitions • connected graph with n vertices and n–1 edges • an edge-maximal acyclic graph • graph with a unique path between any 2 vertices Albert R Meyer, April 8, 2013 tree-def.22 5
MIT OpenCourseWare http://ocw.mit.edu 6.042J / 18.062J Mathematics for Computer Science Spring 20 15 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

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