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Introduction to Graphs

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Graph Definition• Graph : consists of vertices and edges. Each

edge must start and end at a vertex.

• Graph G = (V, E)– V = set of vertices– E = set of edges (VV)

• Example:

www.cs.unc.edu/~plaisted/comp122/19-graph1.ppt

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Graph Definition

• Degree of a vertex : number of edges connected to it

• Example: V4 has degree = 3

V2

V5

V3

V4

V1

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Graph Definition

• Total degree of a graph : sum of the degrees of all the vertices. Note: If a graph has n edges, the total degree = 2n.

• Example: graph has total degree = 10

V2

V3V4

V1

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Graph Definition

• The Handshaking Theorem:– Let G = (V, E). Then

– Proof:• Each edge contributes twice to the degree count of all

vertices.

– Example:• If a graph has 5 vertices, can each vertex have degree

3? Or 4?– The sum is 53 = 15 which is an odd number. Not possible.– The sum is 54 = 20 = 2 |E| and 20/2 = 10 edges. May be possible.

EvVv

2)deg(

www.cs.sfu.ca/~jcliu/MACM101/09-Graph.ppt

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Graph Definition

• Types of graphs– Undirected: edge (u, v) = (v, u); for all v,

» (v, v) E (usually no self loops)

» Some may have self loops.

– Directed: (u, v) is edge from u to v, denoted as u v. • Self loops are allowed.• Is a graph where an edge represents a one-way relation only.

– Cf. undirected graph – an edge represents two-way or symmetric relationship between two vertices.

• The number of directed edges which initiate from vertex v is called the outdegree of v or outdeg(v).

• The number of directed edges which terminate at vertex v is called the indegree of v or indeg(v).

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Representation of Graphs

• Two standard ways– Adjacency Lists.

– Adjacency Matrix.

a

dc

b

a

b

c

d

b

a

d

d c

c

a b

a c

a

dc

b1 2

3 4

1 2 3 41 0 1 1 12 1 0 1 03 1 1 0 14 1 0 1 0

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Adjacency Lists• Consists of an array Adj of |V| lists.• One list per vertex.• For u V, Adj[u] consists of all vertices adjacent to u.

a

dc

b

a

b

c

d

b

c

d

d c

a

dc

b

a

b

c

d

b

a

d

d c

c

a b

a c

If weighted, also store weights in adjacency lists.

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Storage Requirement

• For directed graphs:– Sum of lengths of all adj. lists is

out-degree(v) = |E|

vV

– Total storage: (|V|+|E|)• For undirected graphs:

– Sum of lengths of all adj. lists is

degree(v) = 2|E|

vV

– Total storage: (|V|+|E|)

No. of edges leaving v

No. of edges incident on v. Edge (u,v) is incident on vertices u and v.

www.cs.unc.edu/~plaisted/comp122/19-graph1.ppt

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Pros and Cons: adj list

• Pros– Space-efficient, when a graph is sparse.– Can be modified to support many graph variants.

• Cons– Determining if an edge (u,v) G is not efficient.

• Have to search in u’s adjacency list. (degree(u)) time. (V) in the worst case.

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Adjacency Matrix• |V| |V| matrix A.• Number vertices from 1 to |V| in some arbitrary

manner.• A is then given by:

otherwise0

),( if1],[

EjiajiA ij

a

dc

b1 2

3 4

1 2 3 41 0 1 1 12 0 0 1 03 0 0 0 14 0 0 0 0

a

dc

b1 2

3 4

1 2 3 41 0 1 1 12 1 0 1 03 1 1 0 14 1 0 1 0

A = AT for undirected graphs.

www.cs.unc.edu/~plaisted/comp122/19-graph1.ppt

If weighted, store weights also in adjacency matrix.

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Space and Time

• Space: (|V|2).– Not memory efficient for large graphs.

• Time: to list all vertices adjacent to u: (|V|).• Time: to determine if (u, v) E: (1).• Can store weights instead of bits for weighted

graph.

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Directed Graphs

– Theorem

– Examples (continued):• Adjacency matrix of a directed graph:

– outdeg(V1) = 1, indeg(V1) = 2

– outdeg(V2) = 2, indeg(V2) = 1

– outdeg(V3) = 0, indeg(V3) = 2

– outdeg(V4) = 2, indeg(V4) = 0

VvVv

vvE )outdeg()indeg(

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Each edge e has a weight, wt(e)• graph may be undirected or directed• weight may represent length, cost, capacity, etc• adjacency matrix becomes weight matrix• adjacency lists include weight in node

4

6

6 7

4

5

75

5

8

v w

yx

z

a weighted graph G

www.dcs.gla.ac.uk/~rwi/alg3/Lecture6.ppt

Weighted Graphs

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Vertex cover

• A set of vertices that cover all edges, i.e., at least one vertex of all edges is in the set.

• I.e., set of vertices WV with for all {x,y} E: x W or y W.

• Vertex Cover problem:– Given G, find vertex cover of minimum size

Modified from www.cs.uu.nl/docs/vakken/na/na10-2005b.ppt

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Subset S of vertices such that no two vertices in S are connected

www.cs.rochester.edu/~stefanko/Teaching/06CS282/06-CSC282-17.ppt

Independent Set (I)

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Subset S of vertices such that no two vertices in S are connected

www.cs.rochester.edu/~stefanko/Teaching/06CS282/06-CSC282-17.ppt

Independent Set (II)

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•INSTANCE: graph G•SOLUTION: independent set S in G•MEASURE: maximize the size of S

•INSTANCE: graph G, number K•QUESTION: does G have independent set of size K

OPTIMIZATION VERSION:

DECISION VERSION:

www.cs.rochester.edu/~stefanko/Teaching/06CS282/06-CSC282-17.ppt

Independent Set (III)

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Subset S of vertices such that every two vertices in S

are connected www.cs.rochester.edu/~stefanko/Teaching/06CS282/06-CSC282-17.ppt

Clique (I)

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Clique

INSTANCE: graph G, number KQUESTION: does G have a clique of size K?

www.cs.rochester.edu/~stefanko/Teaching/06CS282/06-CSC282-17.ppt

Clique (II)

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Relations

• The following are equivalent– G has an independent set with at least k

vertices– The complement of G has a clique with at

least k vertices– G has a vertex cover with at most n-k vertices

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Bipartite Graphs

• A bipartite graph G is– a graph in which the vertices V can be partitioned into

two disjoint subsets V1 and V2 such that

• no two vertices in V1 are adjacent;

• no two vertices in V2 are adjacent.

• A complete bipartite graph Km,n is– Is a bipartite graph in which the sets V1 and V2 contain

m and n vertices, respectively, and every vertex in V1 is adjacent to every vertex in V2.

– Q: How many edges ?

http://www.cs.sfu.ca/~jcliu/MACM101/09-Graph.ppt

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Assignment Problem

• Assignment problem.– Input: weighted, complete bipartite graph G =

(L R, E)with |L| = |R|.

– Goal: find a perfect matching of min weight.

1

2

3

4

5

1' 2' 3' 4' 5'

Min cost perfect matching

M = { 1-2', 2-3', 3-5', 4-1', 5-

4' }

cost(M) = 8 + 7 + 10 + 8 + 11

= 44

3 8 9 15 10

4 10 7 16 14

9 13 11 19 10

8 13 12 20 13

1 7 5 11 9www.bioalgorithms.infoAn Introduction to Bioinformatics Algorithms

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Assignment problem and related problems

• L: n jobs, R: m machines, weight of an edge (i,j) in E: time for job i taken on machine j. – Assign n jobs to each of m machines such that the

total processing time is minimized.• Matching: is a subset of E such that no two

edges in the subset are adjacent. – Perfect matching: A matching saturates all vertices.– Maximum matching: A matching with the largest

possible number of edges.• Min-cost maximum matching problem: Find a

max matching in a graph (with edge weights) such that the total edge weight is minimized.