graphs the ultimate data structure - kirkwood …graphs 2 definition of graph • non-linear data...

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Graphs

The ultimate data structure

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Definition of graph

• Non-linear data structure consisting of nodes & links between them (like trees in this sense)

• Unlike trees, graph nodes may be completely unordered, or may be linked in any order suited to the particular application

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Undirected graphs

• Simplest form of graph• Set of nodes (vertices -- one node is a vertex)

and set of links between them (edges)• Either or both sets may be empty• Each edge connects two vertices; as the name

implies, neither vertex is considered to be a point of origin or a point of destination

• An edge may also connect a vertex to itself

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Example graph

v1v0

v4

v2

v3

v1

v0

v2

v3

v4

e0 e0

e1 e1

e2

e2e4

e4

This graph has 5 vertices and4 edges; a graph is defined bywhich vertices are connected, and which edges connect them

This is the same graph,even though its shape isdifferent

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Application: an undirected state graph

• Each vertex represents the state of some object or situation

• Each edge represents the possibility of moving directly from one state to another

• There exists a path from one state to another if there is an edge connecting each vertex and any vertices between them

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Undirected state graph example

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Directed graphs

• Each edge has an orientation, or direction• One vertex connected by the edge is the source, while the other is the target

• Directed graphs are diagrammed like undirected graphs, except the edges are represented by arrows instead of lines

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Directed graph example

a

c

d

e

b

f

Each arrow representsthe direction of theedge; to get frompoint a to pointf, you must passthrough c, d and b(in that order), butto get from f to arequires only goingthrough b (by the shortest route)

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More graph terminology

• Loop: an edge that connects a vertex to itself; in the illustration, edge e3 is a loop

• Multiple edges: two or more edges connecting to vertices in the same direction– e4 and e5 are multiple edges– e1 and e2 are not

• Simple graph: a graph with no loops or multiple edges

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Directed graph application examples

• Route diagrams for airline routes• Course pre- and co-requisites• Program flow charts• Programming language syntax diagrams

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Modeling with graphs - examples

• Niche overlap graph in ecology:– each species represented by vertex– undirected edge connects 2 vertices if the

species compete• Influence graph:

– each person in group is represented by vertex– directed edge from vertex a to vertex b when a

has influence on b

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Modeling with graphs - examples

• Round-robin tournament graph:– each team represented by vertex– (a,b) is an edge if team a beats team b

• Precedence graph:– vertices represent statements in a computer

program– directed edge between vertices means 1st

statement must be executed before 2nd

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Terms related to undirected graphs

• Adjacent: 2 vertices u & v in an undirected graph G are adjacent (neighbors) in G if there is an edge {u,v}

• Incident, connect: if edge e = {u,v}, e is incident with vertices u & v, and e connects u and v

• Endpoints: vertices u & v are endpoints of edge e

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Terms related to undirected graphs

• Degree of vertex in an undirected graph is the number of edges incident with it– loops count twice– degree of vertex v is denoted deg(v)

• Example: what are the degrees of the vertices in this graph?

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Terms related to undirected graphs

• Isolated vertex (like e in previous example) has degree 0, not adjacent to any other vertex

• Pendant vertex (like d in previous example) - adjacent to exactly one vertex

• The sum of the degrees of all vertices in a graph is exactly twice the number of edges in the graph

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Handshaking Theorem

• Let G=(V,E) be an undirected graph with e edges; then 2e = ∑

∈Vvv)deg(

• Note that sum of degrees of vertices is always even; this leads to the theorem:

An undirected graph has an even number of vertices of odd degree

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Terminology related to directed graphs

• Let G be a directed graph, with an edge e=(u,v):– u is initial vertex– v is terminal, or end vertex– u is adjacent TO v– v is adjacent FROM u– initial & terminal vertex of a loop are the same

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Terminology related to digraphs

• In-degree of vertex v, denoted deg-(v), is the number of edges with v as the terminal vertex

• Out-degree of vertex v, denoted deg+(v), is the number of edges with v as the initial vertex

• A loop has one of each

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Terminology related to digraphs

• The sum of in-degrees is equal to the sum of out-degrees

• Both are equal to the number of edges in the graph:

– Let G=(V,E) be a directed graph; then

∑∑∈∈

=+=−VvVv

vv |E|)(deg)(deg

• The undirected graph that results from ignoring arrows in a directed graph is called the underlying undirected graph

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Classes of simple graphs

• Complete graphs: complete graph on n vertices, denoted Kn, is the simple graph that contains exactly one edge between each pair of distinct vertices; Examples:

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Classes of Simple Graphs

• Cycle: Cn, where n 3, consists of n vertices v1, v2, … , vn and edges {v1, v2}, {v2, v3}, … , {vn-1, vn} Examples:

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Classes of Simple Graphs

• Wheel: a cycle with an additional vertex, which is adjacent to all other vertices; example:

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Classes of Simple Graphs

• n-Cube: denoted Qn, is a graph representing the 2n bit strings of length n:

• 2 vertices are adjacent if and only if the bit strings they represent differ in exactly one position. Examples:

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Bipartite Graphs

• A simple graph G is called bipartite if its vertex set V can be partitioned into 2 disjoint non-empty sets V1 and V2 such that:– every edge connects a vertex in V1 with a

vertex in V2– no edge connects 2 vertices in V1 or in V2

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Example: lining up 1st graders

Cate JohnMary MarkMary Ann TerryMary Pat TimMarie Jimmy

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Example - cycle

• C6 is bipartite; can partition its vertex set into 2 distinct sets:– V1 = {v1, v3, v5)– V2 = {v2, v4, v6}– with every edge connecting a

vertex in V1 with one in V2

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ExamplesGraph at left is not bipartite; to divide into2 sets, one set must include 2 vertices and tobe bipartite, those vertices must not be connectedBut every vertex in this graph is connected to2 others

This graph, on the other hand, is bipartite; thetwo sets are V1 = {v1, v3, v5} and V2 = {v2, v4, v6}Note that the definition doesn’t say a vertex inone set can’t connect to more than one vertex inthe other - only that each in one must connect to one in the other, and no vertex in a set can connectto a vertex in the same set

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Examples: are they bipartite?

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Complete Bipartite Graphs

• Denoted Km,n is the graph that has its vertex set partitioned into 2 subsets of m vertices and n vertices

• There is an edge between 2 vertices if and only if one vertex is in the first subset and the other is in the second subset

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Examples

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Applications of Special Types of Graphs: LAN topology

• A star is a complete bipartite K1,n graph:

• A ring is an n-cycle:

• A redundant network may have both a central hub and a ring, forming a wheel:

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Subgraph

• Graph obtained by removing vertices and their associated edges from a larger graph; more formally:

• Subgraph of G=(V,E) is H=(W,F) where W V and F E

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Subgraph Example

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Can combine graphs, forming a union

• Let G1 = (V1, E1) and G2 = (V2, E2)• The union, G1 G2 = (V1 V2, E1

E2)

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Graph implementations

• All of the following representations can be used to create an instance of a directed graph in which loops are allowed:– adjacency list– adjacency matrix – edge lists– edge sets– incidence matrices

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Representing Graphs

• Adjacency list: specifies vertices adjacent with each vertex in a simple graph– can be used for directed graph as well - list

terminal vertices adjacent from each vertexVertex Adjacent vertices a c,d b e c a,d d a,c,e e d,b

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Adjacency matrix• Square grid of true/false values that represent

the edges of a graph• If the intersection of a row and column has a

true value in it, there exists an edge between the vertices represented by the two indexes

• Since this is a directed graph, the value will be true only if there is an edge from the vertex indicated by the row index to the vertex indicated by the column index

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Adjacency matrix

• An adjacency matrix can be stored in a two-dimensional array of bools or ints

• Each vertex is represented by one row and one column

• For a graph with four vertices, the following declaration could be used:

boolean [][] aMatrix = new boolean [4][4];

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Adjacency matrix

[0] [1] [2] [3]

[0]

[1]

[2]

[3]

row

column

The graph above could berepresented by the adjacencymatrix on the right

f t f f

f f f t

f f f f

f t t t

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Representing Graphs: mathematical POV

• Adjacency matrix: n x n zero-one matrix with 1 representing adjacency, 0 representing non-adjacency

• Adjacency matrix of a graph is based on chosen ordering of vertices; for a graph with n vertices, there are n! different adjacency matrices

• Adjacency matrix of graph with few edges is a sparse matrix - many 0s, few 1s

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Simple graph adjacency matrix

With ordering a,b,c,d,e:0 0 1 1 00 0 0 0 11 0 0 1 01 0 1 0 10 1 0 1 0

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Using adjacency matrices for multigraphs and pseudographs

• With multiple edges, matrix is no longer zero-one: if more than one edge exists, list the number of edges

• A loop is represented with a 1 in position i,i• Adjacency matrices for simple graphs,

including multigraphs and pseudographs, are symmetric

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Adjacency matrices and digraphs

• For directed graphs, the adjacency matrix has a 1 in its (i,j)th position if there is an edge from vi to vj

• Directed multigraphs can be represented by placing the number of edges in in position i,j that exist from vi to vj

• Adjacency matrices for digraphs are not necessarily symmetric

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Edge lists

• Linked lists are used to represent vertices• Each vertex is a linked list; nodes in each

list contain numbers indicating the target vertices for that particular source vertex

• In other words, if list A contains a link to vertex B, there is an edge from A to B

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Edge lists

• For n vertices, there must be n lists; often the lists are stored as an array of list head references

• A vertex that is not a source would be represented in the array by a NULL pointer

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Edge lists

The graph above could berepesented by the array of edgelists on the right

[0] [1] [2] [3]

1

null

3

null

null

null

1

3

2

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Edge Sets

• Requires a previously programmed implementation of a set ADT

• Properties of a set:– contains an unordered collection of entries– each item is unique– operations include insertion, deletion, and

checking for the presence of a specific item• To represent a graph with 4 vertices, can

declare an array of 4 sets of ints

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Incidence Matrices

• In an adjacency matrix, both rows and columns represent vertices

• In an incidence matrix, rows represent vertices while columns represent edges– A 1 in any matrix position means the edge in

that column is incident with the vertex in that row

– Multiple edges are represented identical columns depicting the edges

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Incidence Matrix Example

e1 e2 e3 e4 e5 e6 e7 e8 e9a 1 1 1 1 0 0 0 0 0b 0 0 0 0 1 1 0 0 0c 0 1 0 0 0 0 1 1 0d 0 0 0 1 0 0 0 1 1e 1 0 1 0 0 1 0 0 1

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Choosing an implementation

• Adjacency matrix is easier to implement & use than any of the edge-based solutions

• Frequency of certain operations may dictate another solution

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Choosing an implementation

• Both adding and removing edges and checking for the presence of a particular edge are O(N) operations in the worst case for edge lists; may be as low as O(logN) for edge sets, depending on representation

• Edge lists are very efficient for operations that execute one time for each edge with a particular vertex

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Disadvantages of adjacency matrix

• Operations on all edges for a vertex require stepping through entire row (whether or not an entry is an edge) -- this is O(N)

• If each vertex has very few edges, an adjacency matrix is mostly wasted space

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Simple graph ADT implementation: labeled graph

• Operations:– constructor– add/removeEdge– size: returns number of vertices in graph– isEdge: returns true if edge exists between specified

source & target– clone– get/setLabel– neighbors: returns array of neighbors of a specified

vertex

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Simple graph ADT implementation: labeled graph

• Data components:– 2-dimensional int array representing edges– array of vertex labels: each vertex includes data

consisting of an informational label

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Labeled graph class definition

public class Graph implements Cloneable{ private boolean[ ][ ] edges; private Object[ ] labels;

public Graph(int n) // n: # of vertices { edges = new boolean[n][n]; // All values initially false labels = new Object[n]; // All values initially null }

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Labeled graph class definition

public void addEdge(int source, int target) { edges[source][target] = true; }

public void removeEdge(int source, int target) { edges[source][target] = false;}

public boolean isEdge(int source, int target) { return edges[source][target];}

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Labeled graph class definition

public Object getLabel(int vertex) { return labels[vertex];}

public void setLabel(int vertex, Object newLabel) { labels[vertex] = newLabel;}

public int size( ) { return labels.length;}

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Labeled graph class definitionpublic int[ ] neighbors(int vertex) { int i , count=0; int[ ] answer; // First count how many edges have the vertex as their source for (i = 0; i < labels.length; i++) { if (edges[vertex][i]) count++; } // Allocate the array for the answer answer = new int[count];

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Labeled graph class definition

// Fill the array for the answer count = 0; for (i = 0; i < labels.length; i++) { if (edges[vertex][i]) answer[count++] = i; } return answer; }

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Labeled graph class definitionpublic Object clone( ) { // Clone a Graph object. Graph answer; try { answer = (Graph) super.clone( ); } catch (CloneNotSupportedException e) { // This would indicate an internal error in the Java runtime system // because super.clone always works for an Object. throw new InternalError(e.toString( )); } answer.edges = (boolean [ ][ ]) edges.clone( ); answer.labels = (Object [ ]) labels.clone( ); return answer; }

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Graph traversals

• Traversal: visiting all nodes in a data structure

• Can’t apply tree traversal methods to graphs, since nodes in a graph don’t have the same child/parent relationships

• Graph traversal methods utilize subsidiary data structures (queues and stacks) to keep track of vertices to be visited

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Graph traversals

• Purpose of traversal algorithm: starting at a given vertex, process the information at that vertex, then move along an edge to process a neighbor

• When traversal is complete, all vertices that can be reached from the start vertex (but not necessarily all vertices in the graph) have been visited

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Depth-first search

• After processing the initial vertex, a depth-first search moves along an edge to one of the vertex’s neighbors

• Once the second vertex is processed, the search moves along an edge to one of its neighbors

• The process continues to move forward as long as vertices can be reached in this manner

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Depth-first search

• The search always proceeds forward until no further forward moves can be made

• Once forward motion is no longer possible at the current node, the search backs up to the previous vertex and visits any previously unvisited neighbors

• Eventually the search backs up to the original starting node, having visited all nodes that were reachable from this point

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Depth-first search example

v1v2

v3

v4

v5

v0

Depth-first printing routine

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public static void depthFirstPrint(Graph g, int start) { boolean[ ] marked = new boolean [g.size( )]; depthFirstRecurse(g, start, marked); }

// The part of a stack is played here by the system stack;// the depthFirstRecurse method stacks up activation records// with each call

Recursive method for depth-first printing

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public static void depthFirstRecurse(Graph g, int v, boolean[ ] marked) { int[ ] connections = g.neighbors(v); int i, nextNeighbor; marked[v] = true; System.out.println(g.getLabel(v)); // Traverse all the neighbors, looking for unmarked vertices: for (i = 0; i < connections.length; i++) { nextNeighbor = connections[i]; if (!marked[nextNeighbor]) depthFirstRecurse(g, nextNeighbor, marked); } }

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Breadth-first search

• Uses queue to keep track of vertices which might still have unprocessed neighbors

• Initial vertex is processed, marked, and placed in queue

• Repeats the following until queue is empty:– remove vertex from front of queue– for each unmarked neighbor of the current

vertex, mark the neighbor, then place the neighbor on the queue

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Breadth-first search example

v1v2

v3

v4

v5

v0

Queue of visited nodes

v0v1v2v2 v4 v4

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Graph Isomorphism

• The simple graphs G1=(V1,E1) and G2=(V2,E2) are isomorphic if there is a one-to-one and onto function f from V1 to V2 with vertices a and b adjacent in G1 if and only if f(a) and f(b) are adjacent in G2 for all vertices a and b in G1

• Function f is called an isomorphism

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Determining Isomorphism

• The number of vertices, number of edges, and degrees of the vertices are invariants under isomorphism

• If any of these quantities differ in two simple graphs, the graphs are not isomorphic

• If these quantities are the same, the graphs may or may not be isomorphic

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Determining Isomorphism

• To show that a function f from the vertex set of graph G to the vertex set of graph H is an isomorphism, we need to show that f preserves edges

• Adjacency matrices can help with this: we can show that the adjacency matrix of G is the same as the adjacency matrix of H when rows and columns are arranged according to f

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ExampleConsider the two graphs below:

• Both have six vertices and seven edges;• Both have four vertices of degree 2 and two vertices of degree 3;• The subgraphs of G and H consisting of vertices of degree 2 and the edges connecting them are isomorphic

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Example continued

• Need to define a function f and then determine whether or not it is an isomorphism:

– deg(u1) in G is 2; u1 is not adjacent to any other degree 2 vertex, so the image of u1 must be either v4 or v6 in H

– can arbitrarily set f(u1) = v6 (if that doesn’t work, we can try v4)

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Example continued

• Continuing definition of f:– Since u2 is adjacent to u1, the possible images

of u2 are v3 and v5– We set f(u2) = v3 (again, arbitrarily)

• Continuing in this fashion, using adjacency and degree of vertices as guides, we derive f for all vertices in G, as shown on next slide

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Example continued

f(u1) = v6f(u2) = v3f(u3) = v4f(u4) = v5f(u5) = v1f(u6) = v2

Next, we set up adjacency matrices for G and H, arranging the matrix for H so that the images of G’s vertices line upwith their corresponding vertices …

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Example continued

u1 u2 u3 u4 u5 u6u1 0 1 0 1 0 0u2 1 0 1 0 0 1u3 0 1 0 1 0 0u4 1 0 1 0 1 0u5 0 0 0 1 0 1u6 0 1 0 0 1 0

AG

v6 v3 v4 v5 v1 v2v6 0 1 0 1 0 0v3 1 0 1 0 0 1v4 0 1 0 1 0 0v5 1 0 1 0 1 0v1 0 0 0 1 0 1v2 0 1 0 0 1 0

AH

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Example concluded

• Since AG = AH, it follows that f preserves edges, and we conclude that f is an isomorphism; thus G and H are isomorphic

• If f were not an isomorphism, we would not have proved that G and H are not isomorphic, since another function might show isomorphism