Simple Graphs Connectedness Trees Copyright Albert R Meyer

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Simple Graphs: Connectedness, Trees Copyright © Albert R. Meyer, 2007. All rights reserved. March

Simple Graphs: Connectedness, Trees Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 1

Paths & Simple Paths Lemma: The shortest path between two vertices is simple! Suppose

Paths & Simple Paths Lemma: The shortest path between two vertices is simple! Suppose path from u to v crossed itself: Proof: u Copyright © Albert R. Meyer, 2007. All rights reserved. c March 7, 2007 v 2

Paths & Simple Paths Lemma: The shortest path between two vertices is simple! Then

Paths & Simple Paths Lemma: The shortest path between two vertices is simple! Then path without c ···c is shorter: u Copyright © Albert R. Meyer, 2007. All rights reserved. c March 7, 2007 v 3

Connected Graphs A connected graph: there is a path between every two vertices. Copyright

Connected Graphs A connected graph: there is a path between every two vertices. Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 4

Connected Components Every graph consists of separate connected pieces (subgraphs) called connected components Copyright

Connected Components Every graph consists of separate connected pieces (subgraphs) called connected components Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 5

Connected Components 13 10 12 4 26 8 16 66 East Campus E 17

Connected Components 13 10 12 4 26 8 16 66 East Campus E 17 E 25 Med Center Inﬁnite corridor 3 connected components The more connected components, the more “broken up" the graph is. Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 6

Connected Components The connected component of vertex v : Copyright © Albert R. Meyer,

Connected Components The connected component of vertex v : Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 7

Connected Components So a graph is connected iff it has only 1 connected component

Connected Components So a graph is connected iff it has only 1 connected component Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 8

Cycles A cycle is a path that begins and ends with same vertex a

Cycles A cycle is a path that begins and ends with same vertex a v b w path: v ···b ···w ···a ···v also: a ···v ···b ···w ···a Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 9

Cycles A cycle is a path that begins and ends with same vertex a

Cycles A cycle is a path that begins and ends with same vertex a v b w also: a ···w ···b ···v ···a Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 10

Simple Cycles A simple cycle is a cycle that doesn’t cross itself v w

Simple Cycles A simple cycle is a cycle that doesn’t cross itself v w path: v ···w ···v Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 also: w ···v ···w 11

Trees A tree is a connected graph with no cycles. Copyright © Albert R.

Trees A tree is a connected graph with no cycles. Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 12

More Trees Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007

More Trees Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 13

Other Tree Definitions • A tree is a graph with a unique path between

Other Tree Definitions • A tree is a graph with a unique path between any 2 vertices. • A tree is a connected graph with n vertices and n – 1 edges. • A tree is an edge-minimal connected graph. Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 14

Be careful with these definitions Is a tree is a graph with n vertices

Be careful with these definitions Is a tree is a graph with n vertices and n – 1 edges? NO: Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 15

Some trees with five vertices Copyright © Albert R. Meyer, 2007. All rights reserved.

Some trees with five vertices Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 16

Some trees with five vertices Exercise: Prove that all trees with five vertices are

Some trees with five vertices Exercise: Prove that all trees with five vertices are isomorphic to one of these three. Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 17

Spanning Trees A spanning tree: a subgraph that is a tree on all the

Spanning Trees A spanning tree: a subgraph that is a tree on all the vertices. Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 19

Spanning Trees Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007

Spanning Trees Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 20

Spanning Trees a spanning tree Copyright © Albert R. Meyer, 2007. All rights reserved.

Spanning Trees a spanning tree Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 21

Spanning Trees another spanning tree (can have many) Copyright © Albert R. Meyer, 2007.

Spanning Trees another spanning tree (can have many) Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 22

Spanning Trees A spanning tree: a subgraph that is a tree on all the

Spanning Trees A spanning tree: a subgraph that is a tree on all the vertices. Always exists: find minimum edge-size, connected subgraph on all the vertices. Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 23

CONNECTEDNESS An edge is a cut edge if removing it from the graph disconnects

CONNECTEDNESS An edge is a cut edge if removing it from the graph disconnects two vertices. Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 24

Cut Edges Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007

Cut Edges Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 25

Cut Edges B B is a cut edge Copyright © Albert R. Meyer, 2007.

Cut Edges B B is a cut edge Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 26

Cut Edges deleting B gives two components Copyright © Albert R. Meyer, 2007. All

Cut Edges deleting B gives two components Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 27

Cut Edges A A is not a cut edge Copyright © Albert R. Meyer,

Cut Edges A A is not a cut edge Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 28

Cut Edges and Cycles Lemma: An edge is a cut edge iff it is

Cut Edges and Cycles Lemma: An edge is a cut edge iff it is not traversed by a simple cycle. Proof: Team problem. Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 29

Cut Edges Fault-tolerant design: In a tree, every edge is a cut edge (bad)

Cut Edges Fault-tolerant design: In a tree, every edge is a cut edge (bad) In a mesh, no edge is a cut edge (good) Tradeoff edges for failure tolerance Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 30

k-Connectedness Def: k-connected iff remains connected when any k-1 edges are deleted. Copyright ©

k-Connectedness Def: k-connected iff remains connected when any k-1 edges are deleted. Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 31

k-Connectedness Example: is (n-1)-connected Copyright © Albert R. Meyer, 2007. All rights reserved. March

k-Connectedness Example: is (n-1)-connected Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 32

Team Problems 1 3 Copyright © Albert R. Meyer, 2007. All rights reserved. March

Team Problems 1 3 Copyright © Albert R. Meyer, 2007. All rights reserved. March 7, 2007 33

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