1. UMass Lowell Computer Science 91.404 Analysis of Algorithms Prof. Karen Daniels Fall, 2001 Wednesday, 9/26/01 Graph Basics

2. Introductory Graph Concepts ä G= (V,E) ä Vertex Degree ä Self-Loops B E C F D A B E C F D A ä Directed Graph (digraph) ä Degree: in/out ä Self-Loops allowed ä Undirected Graph ä No Self-Loops ä Adjacency is symmetric This treatment follows 91.503 textbook Cormen et al. Some definitions differ slightly from other graph literature.

3. Introductory Graph Concepts: Representations B E C F D A B E C F D A ä Undirected Graph ä Directed Graph (digraph) A B C D E F ABCDEF ABCDEF A BC B ACEF C AB D E E BDF F BE A BC B CEF C D D E BD F E Adjacency Matrix Adjacency List Adjacency Matrix Adjacency List This treatment follows 91.503 textbook Cormen et al. Some definitions differ slightly from other graph literature.

4. Introductory Graph Concepts: Paths, Cycles ä Path: ä length: number of edges ä simple: all vertices distinct ä Cycle: ä Directed Graph: ä forms cycle if v 0 =v k and k>=1 ä simple cycle: v 1,v 2..,v k also distinct ä self-loop is cycle of length 1 ä Undirected Graph: ä forms (simple) cycle if v 0 =v k and k>=3 ä simple cycle: v 1,v 2..,v k also distinct B E C F D A path path B E C F D A simple cycle simple cycle This treatment follows 91.503 textbook Cormen et al. Some definitions differ slightly from other graph literature. B E C F D A simple cycle = simple cycle =

5. Introductory Graph Concepts: Connectivity ä Undirected Graph: connected ä every pair of vertices is connected by a path ä one connected component ä connected components: ä equivalence classes under “is reachable from” relation ä Directed Graph: strongly connected ä every pair of vertices is reachable from each other ä one strongly connected component ä strongly connected components: ä equivalence classes under “mutually reachable” relation B E C F D A B E C F D Aconnected 2 connected components not strongly connected strongly connected component B E C F D A B E C F D A This treatment follows 91.503 textbook Cormen et al. Some definitions differ slightly from other graph literature.