1. Graph Theory

2. The Koenigsberg Bridge Problem
Bridges around the island Kneiphof
The Koenigsberg bridge problem: Starting at some land area (in four areas), is it possible to return to our starting location after walking across each of the seven bridges exactly once?
In 1763, Leonhard Euler solved the problem by using a graph (actually a multigraph) in which the land areas are vertices and the bridges are edges.

3. Euler’s Solution
Euler defined the degree of a vertex as the number of edges incident on it.
He then showed that there is a walk starting at any vertex, going through each edge exactly once, and terminating at the starting vertex iff the degree of each vertex is even.
A walk that does this is called Eulerian.
Graphs have been used in a wide variety of applications, including
analysis of electrical circuits
finding shortest routes
project planning
the identification of chemical compounds
social network analysis.
Graphs may be the most widely used of all mathematical structures.

4. The Graph Abstract Data Type
A graph, G, consists of two sets: a finite, nonempty set of vertices, V(G), and a finite, possibly empty set of edges, E(G).
We may write G=(V, E) to represent a graph.
An undirected graph is one in which the pair of vertices representing any edge is unordered.
E.g. (v0, v1)=(v1, v0)
A directed graph is one in which we represent each edge as a directed pair of vertices.
E.g. <v0, v1> represents an edge in which v0 is the tail and v1 is the head.

5. Sample Graphs 1 2 1 2 1 3 3 4 5 6 G1 2 G2 G3 complete graph1 2 1 2 1 3 3 4 5 6 G1 2 G2
complete graph
incomplete graph G3
V(G1)={0,1,2,3} E(G1)={(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)}
V(G2)={0,1,2,3,4,5,6} E(G2)={(0,1),(0,2),(1,3),(1,4),(2,5),(2,6)}
V(G3)={0,1,2} E(G3)={<0,1>,<1,0>,<1,2>}

6. Complete Graph (Clique)
A complete graph is a graph that has the maximum number of edges
For undirected graph with n vertices, the maximum number of edges is n(n-1)/2
For directed graph with n vertices, the maximum number of edges is n(n-1)
Examples 1 2 1 2 3 3

7. Adjacent and Incident If (v0, v1) is an edge in an undirected graph,
v0 and v1 are adjacent
The edge (v0, v1) is incident on vertices v0 and v1
If <v0, v1> is an edge in a directed graph
v0 is adjacent to v1, and v1 is adjacent from v0
The edge <v0, v1> is incident on v0 and v1
Restrictions on graphs:
A graph may not have an edge from a vertex, i, back to itself. Such edges are known as self loops.
A graph may not have multiple occurrences of the same edge. If we remove this restriction, we obtain a data object referred to as a multigraph.

8. Subgraph
A subgraph of G is a graph G’ such that V(G’)V(G) and E(G’)  E(G).

9. Path, Cycle, and Connected Component
A path from vertex vp to vertex vq in a graph, G, is a sequence of vertices, vp, vi1, vi2, …, vin, vq,such that (vp, vi1), (vi1, vi2), …, (vin, vq) are edges in an undirected graph.
The length of a path is the number of edges on it.
A simple path is a path in which all vertices, except possibly the first and the last, are distinct.
A cycle is a simple path in which the first and the last vertices are the same.
In an undirected graph G, two vertices, v0, v1, are connected if there is a path in G from v0 to v1.
A connected component, or simply a component, of an undirected graph is a maximal connected subgraph.
A tree is a graph that is connected and acyclic.

10. Connected Components in Directed Graph
Weakly Connected Component (WCC)
Ignore directionality and find the connected components
Strongly Connected Component (SCC)
For all node-pairs, i and j, there is a path from node i to node j and there is a path from node j to node i
Rescursively Strongly Connected Component (RSCC)
For all node-pairs, i and j, the path from node i to node j and the path from node j to node i must be the same

11. Degree
The degree of a vertex is the number of edges incident to that vertex
For directed graph,
the in-degree of a vertex v is the number of edges that have v as the head
the out-degree of a vertex v is the number of edges that have v as the tail
If di is the degree of a vertex i in a graph G with n vertices and e edges, the number of edges is 3 2 1 2
in:1, out: 1 3 3 3 1 2 3 1
in: 1, out: 2 3 4 5 6 3 1 3 1 1 1
in: 1, out: 0 2

12. Adjacency Matrix
Let G=(V,E) be a graph with n vertices. The adjacency matrix of G is a two-dimensional n by n array, say adj_mat
If the edge (vi, vj) is in E(G), adj_mat[i][j]=1.
If there is no such edge in E(G), adj_mat[i][j]=0.
The adjacency matrix for an undirected graph is symmetric; the adjacency matrix for a digraph need not be symmetric.
Merits of adjacency matrix
To determine the connection of vertices is easy
The degree of a vertex is
For a digraph, the row sum is the out_degree, while the column sum is the in_degree
Drawback of adjacency matrix
for sparse graph, most of terms in the adjacency matrix equal zero.

13. Examples for Adjacency Matrix1 2 3 4 5 6 7 1 2 3 1 2 G2 G1
symmetric
undirected: n2/2
directed: n2 G4

14. Adjacency Lists
Adjacency lists: replace the n rows of the adjacency matrix with n linked lists, one for each vertex in G.
An undirected graph with n vertices and e edges ==> n head nodes and 2e list nodes.
A directed graph with n vertices and e edges ==> n head nodes and e list nodes.

15. Interesting Operations for Adjacency Lists
Degree of a vertex in an undirected graph
# of nodes in adjacency list
Number of edges in a graph
determined in O(n+e)
Out-degree of a vertex in a directed graph
number of nodes in its adjacency list
In-degree of a vertex in a directed graph
traverse the whole data structure

16. Inverse Adjacency Lists
Determine in-degree of a vertex in a fast way. 
NULL 1 2 1
NULL
NULL 2

17. Lab 1: Representing a Network in UCINET
friendship
spouse
friendship + spouse

18. Homework 1: Drawing a Ring of Ring
5*5:
4*4
4*5
5*4
3*6
6*3 …

19. Weighted Edges
In many applications, the edges of a graph are assigned weights. These weights may represent the distance from one vertex to another or the cost of going from one vertex to an adjacent vertex.
A graph with weighted edges is called a network.
Example
Cost adjacency matrix represents the airline network.

20. Elementary Graph Operations
Recall four kinds of tree traversals:
Preorder
Inorder
Postorder
Level order
To visit all vertices in G that are reachable from v, that is, all vertices that are connected to v.
Depth-First Search (DFS) preorder
Breadth-First Search (BFS) level order
DFS and BFS can be used to find
Connected components
Spanning tree

21. Depth-First Search 1 2 3 4 5 6 7 Major steps: Consider example graph
Visiting the start vertex, v. (that is, printing it)
Select an unvisited vertex, w, from v’s adjacent nodes and carry out a depth first search on w.
Preserve the current position in v’s adjacent nodes by placing it on a stack.
Consider example graph
Traversal order: 0, 1, 3, 7, 4, 5, 2, 6
Time complexity
Adjacency lists: O(e)
Adjacency matrix: O(n2) 1 2 3 4 5 6 7

22. Breadth-First Search 1 2 3 4 5 6 7 Major steps: Consider example graph
Visiting the start vertex, v. (that is, printing it)
Visits each of the vertices on v’s adjacent nodes.
As we visit each vertex, we place the vertex in a queue.
Consider example graph
Traversal order: 0, 1, 2, 3, 4, 5, 6, 7
Time complexity
Adjacency lists: O(e)
Adjacency matrix: O(n2) 1 2 3 4 5 6 7

23. Connected Components in Undirected Graph
In an undirected graph G, two vertices, v0, v1, are connected if there is a path in G from v0 to v1.
A connected component, or simply a component, of an undirected graph is a maximal connected subgraph.
We can find a connected component by either DFS or BFS. Time complexity:
Adjacency lists: O(n + e)
Adjacency matrix: O(n2)

24. Spanning Trees
A spanning tree is any tree that consists solely of edges in G and that includes all the vertices in G.
We may use either DFS or BFS to create a spanning tree.
When DFS is used, a depth first spanning tree is built.
When BFS is used, a breadth first spanning tree is built.
E(G): T (tree edges) + N (nontree edges) where T: set of edges used during search N: set of remaining edges
While adding a nontree edge into any spanning tree, this will create a cycle.

25. DFS vs. BFS Spanning Tree1 2 1 2 1 2 3 4 5 6 3 4 5 6 3 4 5 6 7 7
nontree edge
cycle 7
DFS Spanning
BFS Spanning

26. Minimal Subgraph
A spanning tree is a minimal subgraph, G’, of G such that V(G’)=V(G) and G’ is connected.
Any connected graph with n vertices must have at least n-1 edges.
A biconnected graph is a connected graph that has no articulation points. 1 2 3 4 5 6 7

27. one connected component two connected components 8 9 8 9 7 1 2
a connected graph 3 5 4 6
one connected component
two connected components 7 7 1 1 2 2 5 3 5 3 6 4 6 4

28. Biconnected Components
Biconnected components: A maximal connected subgraph H
no subgraph that is both biconnected and properly contains H 8 9 7 7 1 7 1 7 1 2 2 3 5 3 5 3 5 4 6 4 6
biconnected components

29. Finding the Binconnected Components
Either by examining depth first spanning tree
Or by UCINET…

30. Lab 2: Analyzing Connected Components
Weakly Connected Components
Strongly Connected Components
Recursively Connected Components
Bi-Connected Components

31. Shortest Paths Given a graph and two nodes, A and B, in the graph. V0
Is there a path from A to B?
Is there is more than one path from A to B, which is the shortest path?
Example: shortest paths from v0 to all destinations V0 V4 V1 V5 V3 V2 45
(a)
path length
1) v0 v
2) v0 v2 v
3) v0 v2 v3 v1 45
4) v0 v
(b)

32. Determining the Shortest Paths
Main loops
for (i=0; i<n; i++) {
s[i] = FALSE;
dist[i] = length[v][i]; }
s[v] = TRUE;
dist[v] = 0;
for (i=0; i<n-2; i++) {//determine n-1 paths from v
u = choose(n); //choose returns a value u such that:
//dist[u]=minimum dist[w], where s[w]=FALSE
s[u] = TRUE;
for (w=0; w<n; w++)
if (!s[w])
if (dist[u]+length[u][w]<dist[w])
dist[w] = dist[u]+length[u][w];
Time complexity: O(n2)

33. Example for the Shortest Paths
Boston 4
1500
Chicago
San
Francisco
Denver
250
1200 3
800
1000 1 2
New
York 5
300
1000
1400
New Orleans
1700
900
Los Angeles 7
Miami
1000 6
0 0
Length-adjacency matrix

34. Action of Finding the Shortest Paths
(b)
(c)
(d)
(a)
(f)
(e)
(h)
(g)
(j)
(i)

35. All-Pairs Shortest Paths
Find the shortest paths between all pairs of vertices.
Solution 1
Apply shortest path n times with each vertex as source. O(n3)
Solution 2
Represent the graph G by its length-adjacency matrix with length[i][j]. If the edge <i,j> is not in G, the length[i][j] is set to some sufficiently large number.
Ak[i][j] is the cost of the shortest path form i to j, using only those intermediate vertices with an index <= k.
A-1[i][j]=length[i][j]
Calculate the A0, A1, A2, ..., An-1 from A-1 iteratively.
Ak[i][j]=min{Ak-1[i][j], Ak-1[i][k]+Ak-1[k][j]}, k>=0

36. Examples for All-Pairs Shortest Paths

37. Transitive Closure 1 2 3 4 (a) Digraph G (b) Adjacency matrix A for G
Given a graph with unweighted edges, determine if there is a path from i to j for all i and j.
(1) Require positive path (> 0) lengths.
(2) Require nonnegative path (0) lengths.
transitive closure matrix
reflexive transitive closure matrix 1 2 3 4 1 2 3 4
(a) Digraph G (b) Adjacency matrix A for G 1 2 3 4 1 2 3 4
cycle
reflexive
(c) transitive closure matrix A (d) reflexive transitive closure matrix A*

38. Lab 3: Calculating Shortest Paths
Shortest-path-length matrix
Reflexive transitive closure matrix
Dichotomizing shortest-path-length matrix