discrete mathematics and its...

Discrete Mathematics and Its ApplicationsLecture 7: Graphs: Graph Models and Graph Types

MING GAO

DaSE@ECNU(for course related communications)

mgao@dase.ecnu.edu.cn

Jun. 15, 2020

Outline

1 Graphs

2 Graph Models

3 Graph Terminology and Special Types of GraphsBasic terminologySome Special Simple GraphsNew Graphs from Old

4 Take-aways

MING GAO (DaSE@ECNU) Discrete Mathematics and Its Applications Jun. 15, 2020 2 / 33

Graphs

Graphs - why should we care?

Motivation

Graphs in real world

“YahooWeb graph”: 1B vertices(Web sites), 6B edges (links)

Facebook, Twitter, etc: more than 1B users

Food Web: all biologies, food chain

Power-grid: vertices (plants or consumers), edges (power lines)

Airline route: vertices (airports), edges (flights)

Adoption: users purchase products, adopt services, etc.

Graphs

Motivation questions

Questions

What do real graphs look like?

What properties of vertices, edges are important to model?What local and global properties are important to measure?

Are graphs helpful to understand the real world?

Social influenceRecommendationInformation propagationHuman behaviors

Is a sub-graph “normal” (Water army, fraud detection, spamfiltering, etc)?

How to generate realistic graphs?

How to get a “good” sample of a network?

How to design an efficient algorithm to handle large-scalegraphs?

Graphs

Definition

A graph G = (V ,E ) consists of V , a nonempty set of vertices(or nodes) and E , a set of edges. Each edge has either one or twovertices associated with it, called its endpoints. An edge is said toconnect its endpoints.

The set V of a graph G may be infinite (infinite graph).

In this book we will usually consider only finite graphs.

A graph in which each edge connects two different vertices andwhere no two edges connect the same pair of vertices is calleda simple graph.

Graphs that may have multiple edges connecting the samevertices are called multigraphs.

To model this network we need to include edges that connect avertex to itself, such edges are called loops.

Graphs

Definition

A graph G = (V ,E ) consists of V , a nonempty set of vertices(or nodes) and E , a set of edges. Each edge has either one or twovertices associated with it, called its endpoints. An edge is said toconnect its endpoints.

The set V of a graph G may be infinite (infinite graph).

In this book we will usually consider only finite graphs.

A graph in which each edge connects two different vertices andwhere no two edges connect the same pair of vertices is calleda simple graph.

Graphs that may have multiple edges connecting the samevertices are called multigraphs.

To model this network we need to include edges that connect avertex to itself, such edges are called loops.

Graphs

Directed graphs and graph terminology

Definition

A directed graph (or digraph) (V ,E ) consists of a nonempty setof vertices V and a set of directed edges (or arcs) E . Each directededge is associated with an ordered pair of vertices. The directed edgeassociated with the ordered pair (u, v) is said to start at u and endat v .

Graphs

Directed graphs and graph terminology

Definition

A directed graph (or digraph) (V ,E ) consists of a nonempty setof vertices V and a set of directed edges (or arcs) E . Each directededge is associated with an ordered pair of vertices. The directed edgeassociated with the ordered pair (u, v) is said to start at u and endat v .

Graph Models

Graph models I

Social networks

Graphs are extensively used to model social structures based on d-ifferent kinds of relationships between people or groups of people.These social structures are known as social networks.

We can use a simple graph to representwhether two people know each other.

In studies of group behavior, certain people caninfluence the thinking of others.

The graph is to model that two people arerelated by working together in a particular way.

Graph Models

Graph models I

Social networks

Graph Models

Graph models I

Social networks

Graph Models

Graph models I

Social networks

Graph Models

Graph models II

Communication networks

We can model different communications networks using verticesto represent devices and edges to represent the particular type ofcommunications links of interest.

Graphs can be used to model telephone callsmade in a network, such as a long distancetelephone network.

In Sina Weibo, via observing the retweetbehavior, Guo Meimei scandal cascades in thenetwork.

Graph Models

Graph models II

Graph Models

Graph models II

Graph Models

Graph models III

Information networks

Graphs can be used to model various networks that link particulartypes of information.

Graphs can be used to represent citations indifferent types of documents, includingacademic papers, patents, and legal opinions.

The World Wide Web can be modeled as adirected graph where each Web page isrepresented by a vertex and where an edgestarts at the Web page a and ends at the Webpage b if there is a link on a pointing to b.

Graph Models

Graph models III

Graph Models

Graph models III

Graph Models

Graph models IV

Software design applications

Graph models are useful tools in the design of software. We willbriefly describe two of these models here.

A module dependency graph provides a usefultool for understanding how different modules ofa program interact.

Computer programs can be executed morerapidly by executing certain statementsconcurrently. The dependence of statementscan be represented by a directed graph.

Graph Models

Graph models IV

Graph Models

Graph models IV

Graph Models

Graph models V

Transportation networks

We can use graphs to model many different types of transportationnetworks, including road, air, and rail networks, as well shipping net-works.

The resulting graph will generally be a directedmultigraph, as there may be multiple flightsfrom one airport to some other airport duringthe same day.

In such models, vertices represent intersectionsand edges represent roads.

Graph Models

Graph models V

Graph Models

Graph models V

Graph Models

Graph models VI

Biological networks

Many aspects of the biological sciences can be modeled using graphs.

Each species is represented by a vertex. Anundirected edge connects two vertices if thetwo species represented by these verticescompete, i.e., some of the resources they useare the same.

A protein interaction in a living cell occurswhen two or more proteins in that cell bind toperform a biological function.

Graph Models

Graph models VI

Biological networks

Many aspects of the biological sciences can be modeled using graphs.Each species is represented by a vertex. Anundirected edge connects two vertices if thetwo species represented by these verticescompete, i.e., some of the resources they useare the same.

Graph Models

Graph models VI

Biological networks

Many aspects of the biological sciences can be modeled using graphs.Each species is represented by a vertex. Anundirected edge connects two vertices if thetwo species represented by these verticescompete, i.e., some of the resources they useare the same.

Graph Terminology and Special Types of Graphs Basic terminology

Basic concepts for undirected graphs

Neighbors

Two vertices u and v in an undirected graph G are called adjacent(or neighbors) in G if u and v are endpoints of an edge e of G .

Neighborhood

The set of all neighbors of a vertex v of G = (V ,E ), denoted byN(v), is called the neighborhood of v . If A is a subset of V , wedenote by N(A) the set of all vertices in G that are adjacent to atleast one vertex in A. So, N(A) =

⋃v∈AN(v).

Degree

The degree of a vertex in an undirected graph is # edges incidentwith it, except that a loop at a vertex contributes twice to the degreeof that vertex. The degree of the vertex v is denoted by deg(v).

Neighbors

Neighborhood

⋃v∈AN(v).

Degree

Neighbors

Neighborhood

⋃v∈AN(v).

Degree

Example

What are the degrees and what are the neighborhoods of the verticesin the graph displayed in the figure?

N(a) = {b, d , e},N(b) = {a, b, c , d , e},N(c) = {b},N(d) = {a, b, e},N(e) = {a, b, d}, .

deg(a) = 4

deg(b) = 6

deg(c) = 1

deg(d) = 5

deg(e) = 6.

Some important conclusions in undirected graphs

The Handshaking theorem

Let G = (V ,E ) be an undirected graph with m edges. Then

2m =∑v∈V

deg(v).

(Note that this applies even if multiple edges and loops are present.)

An undirected graph has an even number of vertices of odd degree.Proof: Let V1 and V2 be the set of vertices of even and odd degreesin an undirected graph G = (V ,E ) with m edges. Then

2m =∑v∈V

deg(v) =∑v∈V1

deg(v) +∑v∈V2

deg(v).

Thus, there are an even number of vertices of odd degree.

2m =∑v∈V

deg(v).

2m =∑v∈V

deg(v) =∑v∈V1

deg(v) +∑v∈V2

deg(v).

2m =∑v∈V

deg(v).

2m =∑v∈V

deg(v) =∑v∈V1

deg(v) +∑v∈V2

deg(v).

Basic concepts for directed graphs

Adjacent

When (u, v) is an edge of the graph G with directed edges, u is saidto be adjacent to v and v is said to be adjacent from u. The vertexu is called the initial vertex of (u, v), and v is called the terminal orend vertex of (u, v). The initial vertex and terminal vertex of a loopare the same.

Degree

In a graph with directed edges the in-degree of a vertex v , denoted bydeg−(v), is # edges with v as their terminal vertex. The out-degreeof v , denoted by deg+(v), is # edges with v as their initial vertex.(Note that a loop at a vertex contributes 1 to both the in-degree andthe out-degree of this vertex.)

Basic concepts for directed graphs

Adjacent

When (u, v) is an edge of the graph G with directed edges, u is saidto be adjacent to v and v is said to be adjacent from u. The vertexu is called the initial vertex of (u, v), and v is called the terminal orend vertex of (u, v). The initial vertex and terminal vertex of a loopare the same.

Degree

In a graph with directed edges the in-degree of a vertex v , denoted bydeg−(v), is # edges with v as their terminal vertex. The out-degreeof v , denoted by deg+(v), is # edges with v as their initial vertex.(Note that a loop at a vertex contributes 1 to both the in-degree andthe out-degree of this vertex.)

Degrees on directed graphs

Theorem

Let G = (V ,E ) be a directed graph, Then∑v∈V

deg−(v) =∑v∈V

deg+(v) = |E |.

What are in-degree and out-degree of the vertices in the graph dis-played in the figure?

deg(a)+ = 4, deg(a)− = 2

deg(b)+ = 1, deg(b)− = 2

deg(c)+ = 2, deg(c)− = 3

deg(d)+ = 2, deg(d)− = 2

deg(e)+ = 3, deg(e)− = 3

deg(f )+ = 0, deg(f )− = 0

Degrees on directed graphs

Theorem

Let G = (V ,E ) be a directed graph, Then∑v∈V

deg−(v) =∑v∈V

deg+(v) = |E |.

What are in-degree and out-degree of the vertices in the graph dis-played in the figure?

deg(a)+ = 4, deg(a)− = 2

deg(b)+ = 1, deg(b)− = 2

deg(c)+ = 2, deg(c)− = 3

deg(d)+ = 2, deg(d)− = 2

deg(e)+ = 3, deg(e)− = 3

deg(f )+ = 0, deg(f )− = 0

Graph Terminology and Special Types of Graphs Some Special Simple Graphs

Complete graphs

Definition

A complete graph on n vertices, denoted by Kn, is a simple graphthat contains exactly one edge between each pair of distinct vertices.

A complete graph is also called clique.

A simple graph for which there is at least one pair of distinctvertex not connected by an edge is called non-complete.

Complete graphs

Definition

A complete graph on n vertices, denoted by Kn, is a simple graphthat contains exactly one edge between each pair of distinct vertices.

A complete graph is also called clique.

A simple graph for which there is at least one pair of distinctvertex not connected by an edge is called non-complete.

Cycles and wheels

A cycle Cn, n ≥ 3, consists of n vertices v1, v2, · · · , vn and edges{v1, v2}, {v2, v3}, · · · , {vn−1, vn}, and {vn, v1}.

We obtain a wheel Wn when we add an additional vertex v to a cycleCn, for n ≥ 3, and connect v to each of the n vertices in Cn.

Cycles and wheels

A cycle Cn, n ≥ 3, consists of n vertices v1, v2, · · · , vn and edges{v1, v2}, {v2, v3}, · · · , {vn−1, vn}, and {vn, v1}.

We obtain a wheel Wn when we add an additional vertex v to a cycleCn, for n ≥ 3, and connect v to each of the n vertices in Cn.

n-Cubes

An n-dimensional hypercube, or n-cube, denoted by Qn, is a graphthat has vertices representing the 2n bit strings of length n. Twovertices are adjacent if and only if the bit strings that they representdiffer in exactly one bit position.

We can construct the (n + 1)-cube Qn+1 from Qn by making twocopies of Qn, prefacing the labels on the vertices with a 0 in onecopy of Qn and with a 1 in the other copy of Qn, and adding edgesconnecting two vertices that have labels differing only in the first bit.

n-Cubes

Bipartite graphs

A simple graph G is called bipartite if its vertex set V can be parti-tioned into two disjoint sets V1 and V2 such that every edge in thegraph connects a vertex in V1 and a vertex in V2 (so that no edgein G connects either two vertices in V1 or two vertices in V2). Whenthis condition holds, we call the pair (V1,V2) a bipartition of thevertex set V of G .

C6 is bipartite because itsvertex set can bepartitioned into the twosets V1 = {v1, v3, v5} andV2 = {v2, v4, v6}, andevery edge of C6 connectsa vertex in V1 and a vertexin V2.

Bipartite graphs

A simple graph G is called bipartite if its vertex set V can be parti-tioned into two disjoint sets V1 and V2 such that every edge in thegraph connects a vertex in V1 and a vertex in V2 (so that no edgein G connects either two vertices in V1 or two vertices in V2). Whenthis condition holds, we call the pair (V1,V2) a bipartition of thevertex set V of G .

C6 is bipartite because itsvertex set can bepartitioned into the twosets V1 = {v1, v3, v5} andV2 = {v2, v4, v6}, andevery edge of C6 connectsa vertex in V1 and a vertexin V2.

Examples

Are the graphs G and H displayed in the figure bipartite?

Is there an efficient way to determine whether a graph is a bipartite?

Theorem

A simple graph is bipartite if and only if it is possible to assign oneof two different colors to each vertex of the graph so that no twoadjacent vertices are assigned the same color.

Examples

Are the graphs G and H displayed in the figure bipartite?

Is there an efficient way to determine whether a graph is a bipartite?

Theorem

A simple graph is bipartite if and only if it is possible to assign oneof two different colors to each vertex of the graph so that no twoadjacent vertices are assigned the same color.

Proof.⇒:First, suppose that G = (V ,E ) is a bipartite simple graph. ThenV = V1 ∪ V2, where V1 and V2 are disjoint sets and every edge inE connects a vertex in V1 and a vertex in V2. If we assign one colorto each vertex in V1 and a second color to each vertex in V2, thenno two adjacent vertices are assigned the same color.

⇐:Now suppose that it is possible to assign colors to the vertices ofthe graph using just two colors so that no two adjacent vertices areassigned the same color. Let V1 be the set of vertices assigned onecolor and V2 be the set of vertices assigned the other color. Then,V1 and V2 are disjoint and V = V1 ∪ V2. Furthermore, every edgeconnects a vertex in V1 and a vertex in V2 because no two adjacentvertices are either both in V1 or both in V2. Consequently, G isbipartite.

Proof.⇒:First, suppose that G = (V ,E ) is a bipartite simple graph. ThenV = V1 ∪ V2, where V1 and V2 are disjoint sets and every edge inE connects a vertex in V1 and a vertex in V2. If we assign one colorto each vertex in V1 and a second color to each vertex in V2, thenno two adjacent vertices are assigned the same color.⇐:Now suppose that it is possible to assign colors to the vertices ofthe graph using just two colors so that no two adjacent vertices areassigned the same color. Let V1 be the set of vertices assigned onecolor and V2 be the set of vertices assigned the other color. Then,V1 and V2 are disjoint and V = V1 ∪ V2. Furthermore, every edgeconnects a vertex in V1 and a vertex in V2 because no two adjacentvertices are either both in V1 or both in V2. Consequently, G isbipartite.

Complete bipartite graphs

A complete bipartite graph Km,n is a graph that has its vertex setpartitioned into two subsets of m and n vertices, respectively withan edge between two vertices if and only if one vertex is in the firstsubset and the other vertex is in the second subset.

Complete bipartite graphs

A complete bipartite graph Km,n is a graph that has its vertex setpartitioned into two subsets of m and n vertices, respectively withan edge between two vertices if and only if one vertex is in the firstsubset and the other vertex is in the second subset.

Matching

Definition

A matching M in a simple graph G = (V ,E ) is a subset of the set Eof edges of the graph such that no two edges are incident with thesame vertex.

A maximum matching is a matchingwith the largest number of edges.

A matching M in a bipartite graphG = (V ,E ) with bipartition(V1,V2) is a complete matchingfrom V1 to V2 if every vertex in V1

is the endpoint of an edge in thematching, or equivalently, if|M| = |V1|.

Matching

Definition

A matching M in a simple graph G = (V ,E ) is a subset of the set Eof edges of the graph such that no two edges are incident with thesame vertex.

A maximum matching is a matchingwith the largest number of edges.

A matching M in a bipartite graphG = (V ,E ) with bipartition(V1,V2) is a complete matchingfrom V1 to V2 if every vertex in V1

is the endpoint of an edge in thematching, or equivalently, if|M| = |V1|.

Hall’S Marriage theorem

Theorem

The bipartite graph G = (V ,E ) with bipartition (V1,V2) has a com-plete matching from V1 to V2 if and only if |N(A)| ≥ |A| for allsubsets A of V1.

Proof.⇒:Suppose that there is a complete matching M from V1 to V2.Then, if A ⊂ V1, for every vertex v ∈ A, there is an edge in Mconnecting v to a vertex in V2. Consequently, there are at least asmany vertices in V2 that are neighbors of vertices in V1 as there arevertices in V1. It follows that |N(A)| ≥ |A|.⇐:Basis step: If |V1| = 1, it follows since V1 contains a singlevertex.

Hall’S Marriage theorem

Theorem

The bipartite graph G = (V ,E ) with bipartition (V1,V2) has a com-plete matching from V1 to V2 if and only if |N(A)| ≥ |A| for allsubsets A of V1.

Proof.⇒:Suppose that there is a complete matching M from V1 to V2.Then, if A ⊂ V1, for every vertex v ∈ A, there is an edge in Mconnecting v to a vertex in V2. Consequently, there are at least asmany vertices in V2 that are neighbors of vertices in V1 as there arevertices in V1. It follows that |N(A)| ≥ |A|.⇐:Basis step: If |V1| = 1, it follows since V1 contains a singlevertex.

Proof of Hall’S Marriage theorem

Proof:Inductive step: Let k be a positive integer. If G = (V ,E ) is a bi-partite graph with bipartition (V1,V2), and |V1| = j ≤ k, then thereis a complete matching M from V1 to V2 whenever the conditionthat |N(A)| ≥ |A| for all A ⊂ V1 is met.

Now suppose that H = (W ,F ) is a bipartite graph with bipartition(W1,W2) and |W1| = k + 1.Case I: Suppose ∀j ∈ Z+ with 1 ≤ j ≤ k , the vertices in every subsetof j elements from W1 are adjacent to at least j + 1 elements of W2.Then, we select a vertex v ∈ W1 and an element w ∈ N({v}) with|N({v})| ≥ |{v}| = 1. We delete v and w and all edges incident tothem from H. This produces a bipartite graph H

′with bipartition

(W1 − {v},W2 − {w}). Because |W1 − {v}| = k , the inductivehypothesis tells us there is a complete matching from W1 − {v} toW2−{w}. Adding the edge from v to w to this complete matchingproduces a complete matching from W1 to W2.

Proof:Inductive step: Let k be a positive integer. If G = (V ,E ) is a bi-partite graph with bipartition (V1,V2), and |V1| = j ≤ k, then thereis a complete matching M from V1 to V2 whenever the conditionthat |N(A)| ≥ |A| for all A ⊂ V1 is met.Now suppose that H = (W ,F ) is a bipartite graph with bipartition(W1,W2) and |W1| = k + 1.

Case I: Suppose ∀j ∈ Z+ with 1 ≤ j ≤ k , the vertices in every subsetof j elements from W1 are adjacent to at least j + 1 elements of W2.Then, we select a vertex v ∈ W1 and an element w ∈ N({v}) with|N({v})| ≥ |{v}| = 1. We delete v and w and all edges incident tothem from H. This produces a bipartite graph H

′with bipartition

Proof:Inductive step: Let k be a positive integer. If G = (V ,E ) is a bi-partite graph with bipartition (V1,V2), and |V1| = j ≤ k, then thereis a complete matching M from V1 to V2 whenever the conditionthat |N(A)| ≥ |A| for all A ⊂ V1 is met.Now suppose that H = (W ,F ) is a bipartite graph with bipartition(W1,W2) and |W1| = k + 1.Case I: Suppose ∀j ∈ Z+ with 1 ≤ j ≤ k , the vertices in every subsetof j elements from W1 are adjacent to at least j + 1 elements of W2.Then, we select a vertex v ∈ W1 and an element w ∈ N({v}) with|N({v})| ≥ |{v}| = 1. We delete v and w and all edges incident tothem from H. This produces a bipartite graph H

′with bipartition

Proof of Hall’S Marriage theorem Cont’d

Proof:Case II: Suppose that for some j with 1 ≤ j ≤ k , there is a subsetW

′1 of j vertices such that there are exactly j neighbors of these

vertices in W2. Let W′2 be the set of these neighbors. Then, by the

inductive hypothesis there is a complete matching from W′1 to W

Remove these 2j vertices from W1 and W2 and all incident edges toproduce a bipartite graph K with bipartition (W1 −W

′1,W2 −W

′2).

We will show that the graph K satisfies the condition |N(A)| ≥ |A|for all subsets A of W1 −W

′1. If not, there would be a subset of t

vertices of W1 −W′1 where 1 ≤ t ≤ k + 1− j such that the vertices

in this subset have fewer than t vertices of W2 −W′2 as neighbors.

Then, the set of j + t vertices of W1 consisting of these t verticestogether with the j vertices we removed from W1 has fewer than j+tneighbors in W2, contradicting the hypothesis that |N(A)| ≥ |A| forall A ⊂W1.Hence, we obtain a complete matching from W1 to W2.

′1,W2 −W

′2).

Then, the set of j + t vertices of W1 consisting of these t verticestogether with the j vertices we removed from W1 has fewer than j+tneighbors in W2, contradicting the hypothesis that |N(A)| ≥ |A| forall A ⊂W1.

Hence, we obtain a complete matching from W1 to W2.

′1,W2 −W

′2).

Then, the set of j + t vertices of W1 consisting of these t verticestogether with the j vertices we removed from W1 has fewer than j+tneighbors in W2, contradicting the hypothesis that |N(A)| ≥ |A| forall A ⊂W1.Hence, we obtain a complete matching from W1 to W2.

Graph Terminology and Special Types of Graphs New Graphs from Old

Subgraph

A subgraph of a graphG = (V ,E ) is a graphH = (W ,F ), where W ⊆ V andF ⊆ E . A subgraph H of G is aproper subgraph of G if H = G .

Induced subgraphs

Let G = (V ,E ) be a simple graph. The subgraph induced by asubset W of the vertex set V is the graph (W ,F ), where the edgeset F contains an edge in E if and only if both endpoints of this edgeare in W .

Subgraph

Induced subgraphs

Subgraph

Induced subgraphs

Remove or adding edges of a graph

Remove

Given a graph G = (V ,E ) and an edge e ∈ E , we can produce asubgraph of G by removing the edge e. The resulting subgraph isdenoted G − e = (V ,E − {e}).Similarly, if E

′is a subset of E , we can produce a subgraph of G ,

denoted as (V ,E −E′), by removing the edges in E

′from the graph.

Adding

We can also add an edge e to a graph to produce a new larger graphwhen this edge connects two vertices already in G . We denote byG +e the new graph, denoted as (V ,E ∪{e}), produced by adding anew edge e, connecting two previously non-incident vertices, to thegraph G .

Remove or adding edges of a graph

Remove

Given a graph G = (V ,E ) and an edge e ∈ E , we can produce asubgraph of G by removing the edge e. The resulting subgraph isdenoted G − e = (V ,E − {e}).Similarly, if E

′is a subset of E , we can produce a subgraph of G ,

denoted as (V ,E −E′), by removing the edges in E

′from the graph.

Adding

We can also add an edge e to a graph to produce a new larger graphwhen this edge connects two vertices already in G . We denote byG +e the new graph, denoted as (V ,E ∪{e}), produced by adding anew edge e, connecting two previously non-incident vertices, to thegraph G .

Edge contractions and vertices remove

Edge contractions

An edge contraction which removesan edge e with endpoints u and vand merges u and v into a newsingle vertex w , and for each edge

with u or v as an endpoint replaces the edge with one with w asendpoint in place of u or v and with the same second endpoint.

Vertex removing

When we remove a vertex v and all edges incident to it from G =(V ,E ), we produce a subgraph, denoted by G − v = (V − {v},E ′

),where E

′is the set of edges of G not incident to v .

If V′

is a subset of V , then graph G −V′

is subgraph (V −V′,E

where E′

is the set of edges of G not incident to a vertex in V′.

Edge contractions and vertices remove

Edge contractions

An edge contraction which removesan edge e with endpoints u and vand merges u and v into a newsingle vertex w , and for each edge

with u or v as an endpoint replaces the edge with one with w asendpoint in place of u or v and with the same second endpoint.

Vertex removing

When we remove a vertex v and all edges incident to it from G =(V ,E ), we produce a subgraph, denoted by G − v = (V − {v},E ′

),where E

′is the set of edges of G not incident to v .

If V′

is a subset of V , then graph G −V′

is subgraph (V −V′,E

where E′

is the set of edges of G not incident to a vertex in V′.

Graph union

Definition

The union of two simplegraphs G1 = (V1,E1) andG2 = (V2,E2) is the simplegraph with vertex set V1 ∪ V2

and edge set E1 ∪ E2. Theunion of G1 and G2 is denotedby G1 ∪ G2.

Take-aways

Conclusions

Graphs

Graph models

Graph terminology and special types of graphs

Basic terminologySome special simple graphsNew graphs from old

discrete mathematics and its...

Documents

discrete mathematics and its...

new discrete mathematics and its...

discrete fourier series & discrete fourier...

[introduction] - wordpress.com · · 2012-06-25chapter -...

discrete fourier series & discrete fourier transform

image compression using discrete cosine transform & discrete

fourier transform of discrete time signals discrete … 6...

discrete time systems discrete time systems & difference...

eel3135: discrete-time signals and systems the discrete...

4.1 discrete differential geometry - hao li · 4.1 discrete...

discrete mathematics and its...

discrete mathematics and its...

discrete mathematics and its...

22c:19 discrete math discrete probability

discrete mathematics and its...

discrete surfaces segmentation into discrete planes...

discrete moving frames and discrete integrable...

csi2101 discrete structures:...

discrete mathematics and its...

§1.1 discrete random variables §1.2 discrete random...