Simple Graphs: Connectedness, Trees

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Presentation transcript:

Simple Graphs: Connectedness, Trees

The shortest path between two vertices is simple! Paths & Simple Paths Lemma: The shortest path between two vertices is simple! Proof: Suppose path from u to v crossed itself: u v c

The shortest path between two vertices is simple! Paths & Simple Paths Lemma: The shortest path between two vertices is simple! Then path without c ···c is shorter: c u v

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

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

The more connected components, the more “broken up" the graph is. 4 Inﬁnite corridor 13 10 12 26 8 16 66 E25 Med Center E17 East Campus 3 connected components The more connected components, the more “broken up" the graph is.

The connected component of vertex v : Connected Components The connected component of vertex v :

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

Cycles A cycle is a path that begins and ends with same vertex a v w b path: v ···b ···w ···w ···a ···v also: a ···v ···b ···w ···w ···a

Cycles A cycle is a path that begins and ends with same vertex a v w b also: a ···w ···w ···b ···v ···a

Simple Cycles A simple cycle is a cycle that doesn’t cross itself v w path: v ···w ···v also: w ···v ···w

A tree is a connected graph with no cycles. Trees A tree is a connected graph with no cycles.

More Trees

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.

Be careful with these definitions Is a tree is a graph with n vertices and n – 1 edges? NO:

Some trees with five vertices

Some trees with five vertices Exercise: Prove that all trees with five vertices are isomorphic to one of these three.

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

Spanning Trees

Spanning Trees a spanning tree

Spanning Trees another spanning tree (can have many)

A spanning tree: a subgraph that is a tree on all the vertices. 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.

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

Cut Edges

Cut Edges B B is a cut edge

Cut Edges deleting B gives two components

Cut Edges A A is not a cut edge

edge iff it is not traversed by a simple cycle. Proof: Team problem. Cut Edges and Cycles Lemma: An edge is a cut edge iff it is not traversed by a simple cycle. Proof: Team problem.

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

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

k-Connectedness Example: is (n-1)-connected

Team Problems Problems 13