1. Graph Theory and Algorithm 01
COP 6726: New Directions in Database Systems
Graph Theory and Algorithm 01

2. Basics A graph is a pair G = (V, E) of sets such that E ⊆ [V]2.
The elements of V are the vertices (or nodes, or points) of the Graph G.
The elements of E are its edges (or lines).
A vertex v is incident with an edge e if v ∈ e; then e is an edge at v.

3. Graphs
If all the vertices of G are pairwise adjacent, the G is complete. A complete graph on n vertices is a Kn.
Pairwise non-adjacent vertices or edges are called independent.
A class of graphs that is closed under isomorphism is called a graph property.
If G`⊆ G and G` contains all the edges xy ∈ E with x,y ∈ V, then G` is an induced subgraph of G.
The degree (or valency) dG(v) = d(v) of a vertex v is the number |E(v)| of edges at v.

4. Isomorphism

5. Subgraph A subgraph S of a graph G is a graph such that
The vertices of S are a subset of the vertices of G
The edges of S are a subset of the edges of G
A spanning subgraph of G is a subgraph that contains all the vertices of G
Subgraph
Spanning subgraph

6. Paths A path is a non-empty graph P=(V, E) of the form
V = {x0, x1, …, xk} E = {x0x1, x1x2, …, xk-1xk}
,where the xi are all distinct.
Two a-b paths are independent if and only if a and b are their only common vertices.

7. Cycles
The minimum length of a cycle in a graph G is the girth g(G) of G.
The maximum length of a cycle in G is its circumference.

8. Connectivity
A non-empty graph G is called connected if any two of its vertices are linked by a path in G.
A maximal connected subgraph of G is a component of G. The components are induced subgraphs, and their vertex sets partition V.
If A,B ⊆ V and X ⊆ V ∪ E are such that every AB in G contains a vertex or an edge from X, we say that X separates the sets A and B in G.
A separating set of vertices is a separator.
A vertex which separates two other vertices of the same components is a cut-vertex and an edge separating its ends is a bridge.

9. Non connected graph with two connected components
Connectivity
Connected graph
Non connected graph with two connected components

10. Connectivity
G is called k-connected (for k ∈ ℕ) if G > k and G-X is connected for every set X ⊆ V with |X| < k.
The greatest integer k such that G is k-connected is the connectivity k(G) of G.
The greatest integer l such that G is l-edge connected is the edge-connectivity λ(G) of G.

11. Connectivity

12. Connectivity
Block is a maximal connected subgraph without a cut-vertex. Thus, every block is either a maximal 2-conntected subgraph, or a bridge (with its ends), or an isolated vertex.
Different blocks of G overlap in at most one vertex, which is the a cut-vertex of G.

13. Block Tree

14. Menger’s theorem
Let G=(V,E) be a graph and A,B ⊆ V. Then the minimum number of vertices separating A from B in G is equal to the maximum number of disjoint A-B paths in G.
A graph is k-connected if and only if it contains k independent paths between any two vertices
A graph is k-edge-connected if and only if it contains k edge-disjoint paths between any two vertices.

15. Tree and forest
An acyclic graph, one not contains any cycles, is called a forest. A connected forest is called tree.
The following assentation are equivalent for a graph T:
T is a tree.
Any two vertices of T are linked by a unique path in T.
T is minimally connected.
T is maximally acyclic.
The vertices at distance k from the root have height k and form the kth level of T.

16. Tree and forest
Tree
Forest

17. Tree
A subtree of a graph is a subgraph which is a tree. If this tree is a spanning subgraph, it is called a spanning tree of the graph.

18. Bipartite graphs
A graph G =(V,E) is called r-partite if V admits a partition into r classes such that every edge has its ends in different classes.

19. Matching in bipartite graphs

20. Complete Graph
A complete graph
A complete bipartite graph

21. Contraction and minors

22. Incidence and adjacency matrices
Incidence matrix
Adjacency matrix

23. Basic Graph Algorithm

24. Depth First Search

25. Depth First Search Courtesy : https:// goo.gl/QVPy1i Adjacency Lists
A: F G B: A H C: A D D: C F E: C D G F: E: G: : H: B: I: H: B H C G I D E F
Courtesy : goo.gl/QVPy1i

26. Depth First Search A-F A-G Function call stack: A B H C G I D E F
dfs(A)
A-F A-G
Function call stack:

27. Depth First Search F-E A-F A-G Function call stack: 27 A B H C G I D
visit(F)
F-E E F
dfs(A)
A-F A-G
Function call stack: 27

28. Depth First Search E-C E-D E-G F-E A-F A-G Function call stack: 28 A B
dfs(E)
E-C E-D E-G I D
dfs(F)
F-E E F
dfs(A)
A-F A-G
Function call stack: 28

29. Depth First Search C-A C-D E-C E-D E-G F-E A-F A-G
dfs(C)
C-A C-D B H C G
dfs(E)
E-C E-D E-G I D
dfs(F)
F-E E F
dfs(A)
A-F A-G
Function call stack: 29

30. Depth First Search C-A C-D E-C E-D E-G F-E A-F A-G
dfs(C)
C-A C-D B H C G
dfs(E)
E-C E-D E-G I D
dfs(F)
F-E E F
dfs(A)
A-F A-G
Function call stack: 30

31. Depth First Search D-C D-F C-A C-D E-C E-D E-G F-E A-F A-G
dfs(D)
D-C D-F
dfs(C)
C-A C-D B H C G
dfs(E)
E-C E-D E-G I D
dfs(F)
F-E E F
dfs(A)
A-F A-G
Function call stack: 31

32. Depth First Search D-C D-F C-A C-D E-C E-D E-G F-E A-F A-G
dfs(D)
D-C D-F
dfs(C)
C-A C-D B H C G
dfs(E)
E-C E-D E-G I D
dfs(F)
F-E E F
dfs(A)
A-F A-G
Function call stack: 32

33. Depth First Search D-C D-F C-A C-D E-C E-D E-G F-E A-F A-G
dfs(D)
D-C D-F
dfs(C)
C-A C-D B H C G
dfs(E)
E-C E-D E-G I D
dfs(F)
F-E E F
dfs(A)
A-F A-G
Function call stack: 33

34. Depth First Search C-A C-D E-C E-D E-G F-E A-F A-G
dfs(C)
C-A C-D B H C G
dfs(E)
E-C E-D E-G I D
dfs(F)
F-E E F
dfs(A)
A-F A-G
Function call stack: 34

35. Depth First Search E-C E-D E-G F-E A-F A-G Function call stack: 35 A B
dfs(E)
E-C E-D E-G I D
dfs(F)
F-E E F
dfs(A)
A-F A-G
Function call stack: 35

36. Depth First Search E-C E-D E-G F-E A-F A-G Function call stack: 36 A B
dfs(E)
E-C E-D E-G I D
dfs(F)
F-E E F
dfs(A)
A-F A-G
Function call stack: 36

37. Depth First Search E-C E-D E-G F-E A-F A-G Function call stack: 37 A
dfs(G) B H C G
dfs(E)
E-C E-D E-G I D
dfs(F)
F-E E F
dfs(A)
A-F A-G
Function call stack: 37

38. Depth First Search E-C E-D E-G F-E A-F A-G Function call stack: 38 A B
dfs(E)
E-C E-D E-G I D
dfs(F)
F-E E F
dfs(A)
A-F A-G
Function call stack: 38

39. Depth First Search F-E A-F A-G Function call stack: 39 A B H C G I D
dfs(F)
F-E E F
dfs(A)
A-F A-G
Function call stack: 39

40. Depth First Search A-F A-G Function call stack: 40 A B H C G I D E F
dfs(A)
A-F A-G
Function call stack: 40

41. Depth First Search A-F A-G Function call stack: 41 A B H C G I D E F
dfs(A)
A-F A-G
Function call stack: 41

42. Breadth First Search

43. Breadth First Search A B C D E F G H I
FIFO Queue
Closed Nodes

44. Breadth First Search FIFO Queue Closed Nodes A B C FIFO QueueG H I
FIFO Queue
Closed Nodes
FIFO Queue
Closed Nodes
FIFO Queue
Closed Nodes

45. Dijkstra Algorithm

46. Dijkstra Algorithm 2 1 A B C 1 1 1 1 1 D E F 1 1 3 1 G H I A B C D E F
Priority Queue ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞
Closed Nodes

47. Dijkstra Algorithm 2 1 Priority Queue Closed Nodes A B C D E F G H I AA B C 1 1 1
Priority
Queue
Closed Nodes A B C D E F G H I 1 1 D E F 1 1
Priority
Queue
Closed Nodes A B C D E F G H I 3 1 G H I
Priority
Queue
Closed Nodes A B C D E F G H I

48. Bellman Ford

49. Bellman-Ford Algorithm2 -1 A B C 1 1 1 1 1 D E F 1 1 -3 1 G H I A B C D E F G H I ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

50. Bellman-Ford Algorithm2 -1 A B C D E F G H I A B C 1 1 1 1 1 D E F 1 1 -3 1 G H I

51. Potential Function

52. Potential Function 1 1 -2 1 A B C D

53. Potential Function d ( u, v) = d (u, v) – p(v) + p (u) 1 1 -1 1 -2 1 A1 -1 1 -2 1 A B C D

54. Potential Function
d ( u, v) = d (u, v) – p(v) + p (u) 1 1 -1 A B C D

55. Take Home Message
Basic Graph theory
Basic Graph Algorithms