lecture17: graph i bohyung han cse, postech [email protected] csed233: data structures (2014f)

Lecture17: Graph I

Bohyung HanCSE, POSTECH

[email protected]

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.


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


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


Applications

ImageNet

Gene network


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


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 ) }


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 ) }


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


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


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


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


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


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


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


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


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


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

lecture17: graph i bohyung han cse, postech [email protected] csed233: data structures (2014f)

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