1. Rosen 5th ed., ch. 8.1~8.5 ~44 slides (more later), ~3 lectures
Module #19: Graph Theory
Rosen 5th ed., ch. 8.1~8.5
~44 slides (more later), ~3 lectures
2019/2/19
(c) , Michael P. Frank

2. What are Graphs? Not Our Meaning
General meaning in everyday math: A plot or chart of numerical data using a coordinate system.
Technical meaning in discrete mathematics: A particular class of discrete structures (to be defined) that is useful for representing relations and has a convenient webby-looking graphical representation.
Not Our
Meaning
2019/2/19
(c) , Michael P. Frank

3. Applications of Graphs
Potentially anything (graphs can represent relations, relations can describe the extension of any predicate).
Apps. in networking, scheduling, flow optimization, circuit design, path planning.
Geneology analysis, computer game-playing, program compilation, object-oriented design, …
2019/2/19
(c) , Michael P. Frank

4. Visual Representation of a Simple Graph
Simple Graphs
Correspond to symmetric binary relations R.
A simple graph G=(V,E) consists of:
a set V of vertices or nodes (V corresponds to the universe of the relation R),
a set E of edges / arcs / links: unordered pairs of [distinct?] elements u,v  V, such that uRv.
Visual Representation of a Simple Graph
2019/2/19
(c) , Michael P. Frank

5. Example of a Simple Graph
Let V be the set of states in the far-southeastern U.S.:
V={FL, GA, AL, MS, LA, SC, TN, NC}
Let E={{u,v}|u adjoins v}
={{FL,GA},{FL,AL},{FL,MS}, {FL,LA},{GA,AL},{AL,MS}, {MS,LA},{GA,SC},{GA,TN}, {SC,NC},{NC,TN},{MS,TN}, {MS,AL}} NC TN MS AL SC GA LA FL
2019/2/19
(c) , Michael P. Frank

6. Multigraphs
Like simple graphs, but there may be more than one edge connecting two given nodes.
A multigraph G=(V, E, f ) consists of a set V of vertices, a set E of edges (as primitive objects), and a function f:E{{u,v}|u,vV  uv}.
E.g., nodes are cities, edges are segments of major highways.
Parallel edges
2019/2/19
(c) , Michael P. Frank

7. Pseudographs
Like a multigraph, but edges connecting a node to itself are allowed.
A pseudograph G=(V, E, f ) where f:E{{u,v}|u,vV}. Edge eE is a loop if f(e)={u,u}={u}.
E.g., nodes are campsites in a state park, edges are hiking trails through the woods.
2019/2/19
(c) , Michael P. Frank

8. Directed Graphs
Correspond to arbitrary binary relations R, which need not be symmetric.
A directed graph (V,E) consists of a set of vertices V and a binary relation E on V.
E.g.: V = people, E={(x,y) | x loves y}
2019/2/19
(c) , Michael P. Frank

9. Directed Multigraphs
Like directed graphs, but there may be more than one arc from a node to another.
A directed multigraph G=(V, E, f ) consists of a set V of vertices, a set E of edges, and a function f:EVV.
E.g., V=web pages, E=hyperlinks. The WWW is a directed multigraph...
2019/2/19
(c) , Michael P. Frank

10. Types of Graphs: Summary
Summary of the book’s definitions.
Keep in mind this terminology is not fully standardized...
2019/2/19
(c) , Michael P. Frank

11. §8.2: Graph Terminology
Adjacent, connects, endpoints, degree, initial, terminal, in-degree, out-degree, complete, cycles, wheels, n-cubes, bipartite, subgraph, union.
2019/2/19
(c) , Michael P. Frank

12. Adjacency
Let G be an undirected graph with edge set E. Let eE be (or map to) the pair {u,v}. Then we say:
u, v are adjacent / neighbors / connected.
Edge e is incident with vertices u and v.
Edge e connects u and v.
Vertices u and v are endpoints of edge e.
2019/2/19
(c) , Michael P. Frank

13. Degree of a Vertex Let G be an undirected graph, vV a vertex.
The degree of v, deg(v), is its number of incident edges. (Except that any self-loops are counted twice.)
A vertex with degree 0 is isolated.
A vertex of degree 1 is pendant.
2019/2/19
(c) , Michael P. Frank

14. Handshaking Theorem
Let G be an undirected (simple, multi-, or pseudo-) graph with vertex set V and edge set E. Then
Corollary: Any undirected graph has an even number of vertices of odd degree.
2019/2/19
(c) , Michael P. Frank

15. Directed Adjacency
Let G be a directed (possibly multi-) graph, and let e be an edge of G that is (or maps to) (u,v). Then we say:
u is adjacent to v, v is adjacent from u
e comes from u, e goes to v.
e connects u to v, e goes from u to v
the initial vertex of e is u
the terminal vertex of e is v
2019/2/19
(c) , Michael P. Frank

16. Directed Degree Let G be a directed graph, v a vertex of G.
The in-degree of v, deg(v), is the number of edges going to v.
The out-degree of v, deg(v), is the number of edges coming from v.
The degree of v, deg(v)deg(v)+deg(v), is the sum of v’s in-degree and out-degree.
2019/2/19
(c) , Michael P. Frank

17. Directed Handshaking Theorem
Let G be a directed (possibly multi-) graph with vertex set V and edge set E. Then:
Note that the degree of a node is unchanged by whether we consider its edges to be directed or undirected.
2019/2/19
(c) , Michael P. Frank

18. Special Graph Structures
Special cases of undirected graph structures:
Complete graphs Kn
Cycles Cn
Wheels Wn
n-Cubes Qn
Bipartite graphs
Complete bipartite graphs Km,n
2019/2/19
(c) , Michael P. Frank

19. Complete Graphs
For any nN, a complete graph on n vertices, Kn, is a simple graph with n nodes in which every node is adjacent to every other node: u,vV: uv{u,v}E. K1 K4 K2 K3 K5 K6
Note that Kn has edges.
2019/2/19
(c) , Michael P. Frank

20. How many edges are there in Cn?
Cycles
For any n3, a cycle on n vertices, Cn, is a simple graph where V={v1,v2,… ,vn} and E={{v1,v2},{v2,v3},…,{vn1,vn},{vn,v1}}. C3 C4 C5 C6 C8 C7
How many edges are there in Cn?
2019/2/19
(c) , Michael P. Frank

21. How many edges are there in Wn?
Wheels
For any n3, a wheel Wn, is a simple graph obtained by taking the cycle Cn and adding one extra vertex vhub and n extra edges {{vhub,v1}, {vhub,v2},…,{vhub,vn}}. W3 W4 W5 W6 W8 W7
How many edges are there in Wn?
2019/2/19
(c) , Michael P. Frank

22. Number of vertices: 2n. Number of edges:Exercise to try!
n-cubes (hypercubes)
For any nN, the hypercube Qn is a simple graph consisting of two copies of Qn-1 connected together at corresponding nodes. Q0 has 1 node. Q0 Q1 Q2 Q4 Q3
Number of vertices: 2n. Number of edges:Exercise to try!
2019/2/19
(c) , Michael P. Frank

23. n-cubes (hypercubes)
For any nN, the hypercube Qn can be defined recursively as follows:
Q0={{v0},} (one node and no edges)
For any nN, if Qn=(V,E), where V={v1,…,va} and E={e1,…,eb}, then Qn+1=(V{v1´,…,va´}, E{e1´,…,eb´}{{v1,v1´},{v2,v2´},…, {va,va´}}) where v1´,…,va´ are new vertices, and where if ei={vj,vk} then ei´={vj´,vk´}.
2019/2/19
(c) , Michael P. Frank

24. Bipartite Graphs
A simple graph G=(V, E) is called bipartite if V can be partitioned into V1 and V2 such that every edge connects a vertex in V1 and a vertex in V2.
2019/2/19
(c) , Michael P. Frank

25. Some Applications LAN(Local Area Network)
Star (Hub, Dumb Switch)
Ring(Token Ring)
Hybrid(?)
Interconnection Networks for Parallel Computation
Linear array, need number of hops sometimes
Mesh
HyperCube
2019/2/19
(c) , Michael P. Frank

26. Subgraphs
A subgraph of a graph G=(V,E) is a graph H=(W,F) where WV and FE. G H
2019/2/19
(c) , Michael P. Frank

27. Graph Unions
The union G1G2 of two simple graphs G1=(V1, E1) and G2=(V2,E2) is the simple graph (V1V2, E1E2).
2019/2/19
(c) , Michael P. Frank

28. §8.3: Graph Representations & Isomorphism
Adjacency lists.
Adjacency matrices.
Incidence matrices.
Graph isomorphism:
Two graphs are isomorphic iff they are identical except for their node names.
2019/2/19
(c) , Michael P. Frank

29. Adjacency Lists
A table with 1 row per vertex, listing its adjacent vertices. b a d c e f
2019/2/19
(c) , Michael P. Frank

30. Directed Adjacency Lists
1 row per node, listing the terminal nodes of each edge incident from that node.
2019/2/19
(c) , Michael P. Frank

31. Adjacency Matrices
Matrix A=[aij], where aij is 1 if {vi, vj} is an edge of G, 0 otherwise.
2019/2/19
(c) , Michael P. Frank

32. Graph Isomorphism Formal definition:
Simple graphs G1=(V1, E1) and G2=(V2, E2) are isomorphic if there is a bijection f:V1V2 such that a,bV1, a and b are adjacent in G1 iff f(a) and f(b) are adjacent in G2.
f is the “renaming” function that makes the two graphs identical.
Definition can easily be extended to other types of graphs.
2019/2/19
(c) , Michael P. Frank

33. Determine Isomorphism
It is often difficult! Need O(n!) .
However, G1=(V1, E1) to be isomorphic to G2=(V2, E2):
|V1|=|V2|, |E1|=|E2|.
i.e. they must be invariant.
2019/2/19
(c) , Michael P. Frank

34. Isomorphism Example
If isomorphic, label the 2nd graph to show the isomorphism, else identify difference. d b a b a d c e e c f f
2019/2/19
(c) , Michael P. Frank

35. Are These Isomorphic?
If isomorphic, label the 2nd graph to show the isomorphism, else identify difference.
* Same # of
vertices a b
* Same # of
edges
* Different
# of verts of
degree 2! (1 vs 3) d e c
2019/2/19
(c) , Michael P. Frank

36. §8.4: Connectivity
In an undirected graph, a path of length n from u to v is a sequence of adjacent edges going from vertex u to vertex v.
A path is a circuit if u=v and n>0.
A path traverses the vertices along it.
A path is simple if it contains no edge more than once.
2019/2/19
(c) , Michael P. Frank

37. Paths in Directed Graphs
Same as in undirected graphs, but the path must go in the direction of the arrows.
2019/2/19
(c) , Michael P. Frank

38. Connectedness
An undirected graph is connected iff there is a path between every pair of distinct vertices in the graph.
Theorem1: There is a simple path between any pair of vertices in a connected undirected graph.
Connected components: disjoint connected subgraphs.
The removal of a cut vertex or cut edge separates 1 connected component into 2 if removed.
2019/2/19
(c) , Michael P. Frank

39. Directed Connectedness
A directed graph is strongly connected iff there is a directed path from a to b and from b to a for any two vertices a and b.
It is weakly connected if and only if the underlying undirected graph (i.e., with edge directions removed) is connected.
Any strongly connected graph is weakly connected.
2019/2/19
(c) , Michael P. Frank

40. Paths & Isomorphism
Note that connectedness, and the existence of a circuit or simple circuit of length k are graph invariants with respect to isomorphism.
2019/2/19
(c) , Michael P. Frank

41. Counting Paths w Adjacency Matrices
Let A be the adjacency matrix of graph G.
The number of paths of length k from vi to vj is equal to (Ak)i,j. (The notation (M)i,j denotes mi,j where [mi,j] = M.)
2019/2/19
(c) , Michael P. Frank

42. §8.5: Euler & Hamilton Paths
An Euler circuit in a graph G is a simple circuit containing every edge of G.
An Euler path in G is a simple path containing every edge of G.
A Hamilton circuit is a circuit that traverses each vertex in G exactly once.
A Hamilton path is a path that traverses each vertex in G exactly once.
2019/2/19
(c) , Michael P. Frank

43. Some Useful Theorems
A connected multigraph has an Euler circuit iff each vertex has even degree.
A connected multigraph has an Euler path (but not an Euler circuit) iff it has exactly 2 vertices of odd degree.
If (but not only if) G is connected, simple, has n3 vertices, and v deg(v)n/2, then G has a Hamilton circuit.
2019/2/19
(c) , Michael P. Frank

44. § 8.6:Shortest-Path Problem
A weighted graph can be used to model an airline system or a computer network.
2 problems to be discussed: shortest-path problem and traveling salesman problem.
Algorithm 1 (Dijstra’s Algorithm) finds the length of a shortest path between two vertices in a connected simple undirected weighted graph.
2019/2/19
(c) , Michael P. Frank

45. Dijkstra’s Algorithm G has vertices v0,v1,…vn and weights w(vi,vj)
Procedure Dijkstra(G: weighted connected simple graph)
for i:=1 to n L(vi):=∞ ;
L(v0)=0 ;
S:=Φ;
While
Begin
u:= a vertex not in S with L(u) minimal;
S:=S∪{u};
for all vertices v not in S
if L(u)+w(u,v)<L(v) then L(v):=L(u)+w(u,v)
end
G has vertices v0,v1,…vn and weights w(vi,vj)
S: set of selected vertices on shortest path
2019/2/19
(c) , Michael P. Frank

46. § 8.6:Shortest-Path Problem
The traveling salesman problem asks for the circuit of minimum total weight in a weighted, complete, undirected graph that visits each vertex exactly once and returns to its starting point.
It’s a NP-complete problem
Many approximation algorithm are devised.
2019/2/19
(c) , Michael P. Frank

47. § 8.7:Planar Graph
A graph is called planar if it can be drawn in the plane without any edges crossing.
K4,Q3 are planar while K3,3 is not planar.
Euler’s formula: Let G be a connected planar graph with e edges and v vertices. Let r be the number of regions in a planar representation of G. r=e-v+2.
2019/2/19
(c) , Michael P. Frank

48. Proof of Euler’s Formula (1)
Basis: Since e1=1, v1=2, r1=1, r1=e1-v1+2 is true.
Inductive Step: Assume rn=en-vn+2 is true. Let {an+1,bn+1} be the edge that is added to Gn to obtain Gn+1.
Case I: Both an+1 and bn+1 are already in Gn .(pp.607Fig10(a))
an+1 and bn+1 must be on the boundary of a common region R,
or else it would be impossible to add the edges {an+1,bn+1} .
The addition of this new edge splits R into 2 regions. Consequently, rn+1= rn+1, en+1= en+1, vn+1= vn . Thus, rn+1=en+1-vn+1+2.
2019/2/19
(c) , Michael P. Frank

49. Proof of Euler’s Formula (2)
Case II: an+1 is in Gn and bn+1 is not. (Fig10(b))
The addition of {an+1,bn+1} does not produce new regions, since bn+1 must be in a region that has an+1 on its boundary.
Consequently, rn+1= rn, en+1= en+1, vn+1= vn+1 . Thus, rn+1=en+1-vn+1+2.
Case III: an+1 and bn+1 are not in Gn .
Impossible. (Because Gn+1 must be connected.)
2019/2/19
(c) , Michael P. Frank

50. Other Theorems
Corollary 1 If G is a connected planar simple graph with e edges and v vertices where v  3, then e  3v-6.
Corollary 2 If G is a connected planar simple graph then G has a vertex of degree not exceeding five.
Proof:
2019/2/19
(c) , Michael P. Frank

51. Other Theorems
Corollary 3 If a connected planar simple graph has e edges and v vertices with v3 and no circuits of length three, then e  2v-4.
2019/2/19
(c) , Michael P. Frank

52. Kuratowski’s Theorem
Theorem 2 A graph is nonplanar if and only if it contains a subgraph homomorphic to K3,3 or K5.
2019/2/19
(c) , Michael P. Frank