lecture17: graph i bohyung han cse, postech bhhan@postech.ac.kr csed233: data structures (2014f)

Lecture17: Graph I

Bohyung HanCSE, POSTECH

bhhan@postech.ac.kr

CSED233: Data Structures (2014F)

2 CSED233: Data Structuresby Prof. Bohyung Han, Fall 2014

Graph

• Definition A graph consists of a finite set of vertices, , and a finite

set of edges, .

• Properties and are sets: each vertex

and each edge are unique. Edge: where Vertices and edges may store

elements.

3 CSED233: Data Structuresby Prof. Bohyung Han, Fall 2014

Graphs

• Example: A vertex represents an airport and stores the three letter airport code.‐ An edge represents a flight route between two airports and stores the

mileage of the route.

ORD

PVD

MIA

DFW

SFO

LAX

LGA

HNL

849

802

13871743

1843

1099

1120

1233

337

2555

142

4 CSED233: Data Structuresby Prof. Bohyung Han, Fall 2014

Applications

• Electronic circuits Printed circuit board Integrated circuit

• Transportation networks Highway network Flight network

• Computer networks Local area network Internet Web

• Databases Entity relationship diagram‐

John

DavidPaul

brown.edu

cox.net

cs.brown.edu

att.netqwest.net

math.brown.edu

cslab1bcslab1a

5 CSED233: Data Structuresby Prof. Bohyung Han, Fall 2014

Applications

ImageNet

Gene network

6 CSED233: Data Structuresby Prof. Bohyung Han, Fall 2014

Basic Definitions in Graph

• Undirected graph Graph with undirected edges Undirected edges:

• Directed graph (= digraph) Graph with directed edges Directed edges

• Ordered pair of vertices, • The first vertex is the origin and the second vertex is the destination.

• Sparse graph Graph with few edges , where denotes the number of elements in a set

• Dense graph Graph with many edges:

ORD PVDflight

AA 1206

ORD PVD849

miles

7 CSED233: Data Structuresby Prof. Bohyung Han, Fall 2014

Undirected Graph

𝒙𝒖

𝒗

𝒘 𝒚

𝒙𝒖

𝒗

𝒘 𝒚

𝑉={𝑢 ,𝑣 ,𝑤 , 𝑥 , 𝑦 }𝐸={ (𝑢 ,𝑣 ) , (𝑣 ,𝑥 ) , (𝑢 ,𝑤 ) , (𝑤 ,𝑥 ) , (𝑤 , 𝑦 ) , (𝑥 , 𝑦 ) }

𝑉={𝑢 ,𝑣 ,𝑤 , 𝑥 , 𝑦 }𝐸={ (𝑢 ,𝑣 ) , (𝑤 ,𝑥 ) , (𝑤 , 𝑦 ) , (𝑥 , 𝑦 ) }

8 CSED233: Data Structuresby Prof. Bohyung Han, Fall 2014

Directed Graph

𝒙𝒖

𝒗

𝒘 𝒚

𝑉={𝑢 ,𝑣 ,𝑤 , 𝑥 , 𝑦 }𝐸={ (𝑣 ,𝑢 ) , (𝑤 ,𝑢) , (𝑣 , 𝑥 ) , (𝑥 ,𝑤 ) , (𝑦 ,𝑤 ) , (𝑥 , 𝑦 ) , (𝑦 ,𝑥 ) }

9 CSED233: Data Structuresby Prof. Bohyung Han, Fall 2014

Terminology

• End vertices (or endpoints) of an edge and are the endpoints of

• Edges incident on a vertex , and are incident on .

• Adjacent vertices

Vertices and are adjacent.

• Degree of a vertex Number of incident edges

• Self loop‐ Edge between the same vertex is a self loop.‐

𝒙𝒖

𝒗

𝒘

𝒛

𝒚

𝒆𝒂

𝒆𝒄

𝒆𝒃

𝒆𝒆

𝒆𝒅

𝒆 𝒇

𝒆𝒈

𝒆𝒉

𝒆𝒊

𝒆 𝒋

10 CSED233: Data Structuresby Prof. Bohyung Han, Fall 2014

Terminology (cont’d)

• Path Sequence of vertices

such that Length of a path

• Number of edges on the path• The length of the path from a

vertex to itself is 0.

• Simple path Path such that all its vertices and

edges are distinct Example: Non simple path: ‐

𝒑𝟏

𝒙𝒖

𝒗

𝒘

𝒛

𝒚

𝒑𝟐

11 CSED233: Data Structuresby Prof. Bohyung Han, Fall 2014

Terminology (cont’d)

• Cycle Sequence of vertices

such that and No backtracking

• Simple cycle Cycle such that all its vertices and

edges are distinct. Example: Non simple cycle example: visiting the ‐

same node multiple times

𝒙𝒖

𝒗

𝒘

𝒛

𝒚

𝒄𝟏

𝒄𝟐

12 CSED233: Data Structuresby Prof. Bohyung Han, Fall 2014

Terminology in Directed Graph

• Adjacency For directed graphs, vertex is adjacent

to if and only if .

• Degree Indegree

• Number of incoming edges

Outdegree• Number of outgoing edges

• Path and cycle in directed graph Should consider the direction of edges

• Directed Acyclic Graph (DAG): directed graph with no cycles

𝒙𝒖

𝒗

𝒘

𝒛

𝒚

13 CSED233: Data Structuresby Prof. Bohyung Han, Fall 2014

Properties of Graphs

• For undirected graph , where and

• Each edge is counted twice. with no self loops ‐

and no multiple edges Each vertex has degree at most

• What is the bound for a directed graph?

𝑛=4 ,𝑚=6 , deg(𝑣)=3

14 CSED233: Data Structuresby Prof. Bohyung Han, Fall 2014

Connectivity

• Connected graph There is a path from every vertex to

every other vertex.

• Connected component Maximal connected subgraph

• Complete graph There is an edge between every pair

of vertices.

Connected graph

Non connected graph with two connected components

Complete graph

15 CSED233: Data Structuresby Prof. Bohyung Han, Fall 2014

Connectivity in Directed Graph

• Strongly connected graph There is a path from every vertex to

every other vertex.

• Weakly connected graph There is a path from every vertex to

every other vertex, disregarding the direction of the edges.

Strongly connected graph

Weakly connected graph

16 CSED233: Data Structuresby Prof. Bohyung Han, Fall 2014

Connectivity

• Subgraph A graph is a subgraph of if and

only if and

• A spanning subgraph A subgraph that contains all the

vertices of the original graph

Subgraph

Spanning subgraph

17 CSED233: Data Structuresby Prof. Bohyung Han, Fall 2014

Trees and Forests

• A (free) tree is an undirected graph T such that T is connected T has no cycles This definition of tree is different

from the one of a rooted tree

• Forest A forest is an undirected graph

without cycles The connected components of

a forest are trees

Tree

Forest

18 CSED233: Data Structuresby Prof. Bohyung Han, Fall 2014

Spanning Trees and Forests

• Spanning tree A spanning subgraph that is a tree Not unique unless the graph is a tree

• Spanning forest A spanning subgraph that is a forest

Graph Spanning tree

19