1. Discrete Math II
Howon Kim

2. Agenda
11.1 Definitions
11.2 Subgraphs, Complements, & Graph Isomorphism
11.3 Vertex Degree: Euler Trails & Circuits
11.4 Planar Graphs
Next…

3. Review: Theorem 11.2 Total sum of degrees
If G=(V,E) is an undirected graph or multigraph, then
Proof : As we consider each edge {a,b} in graph G, we find that the edge contributes a count of 1 to each of deg(a), deg(b), and consequently a count of 2 to Thus accounts for deg(v) e7 c e d a b e1 e5 e4 e2 e6 e3
deg(a) = 3
deg(b) = 3
deg(c) = 1
deg(d) = 2
deg(e) = 5
x # of edges
14 = 2|E|

4. Review
Corollary 11.1
For any undirected graph or multigraph, # of vertices of odd degree must be even.
( Proof )
The total sum of degrees is even since 2|E|.
Two kinds of vertices
Even degree : deg ( veven ) should be even number
Odd degree : deg ( vodd ) should be even number
Therefore, the number of odd degree vertices must be even.

5. Review: Euler Circuit
Let G an undirected graph with no isolated vertices.
Euler circuit
a circuit in G that traverses every edge of G exactly once
Euler trail
an open trail from a vertex to another vertex in G, which traverses each edge in G exactly once c d a b

6. Review: Existence of Euler trail Existence of Euler circuit
Let G=(V,E) is an undirected graph with no isolated vertices.
G has an Euler circuit.
G is connected and
every vertex in G has even degree.
Existence of Euler trail
If G=(V,E) is an undirected graph with no isolated vertices, then we can construct an Euler trail.
G is connected and exactly two vertices of odd degrees

7. Review: Isomorphism(동형)
Graph Isomorphism
Let G1=(V1, E1) and G2=(V2, E2) be two graphs.
A function f : V1  V2 is called a graph isomorphism if
(a) f is one-to-one and onto and (전단사)
(b) for all a,b  V1, {a,b}  E1  {f(a), f(b)}  E2
When such a function exists, G1 and G2 are called isomorphic graphs.
In Greek:
ison : “equal”
Morphe “shape”

8. Review: k-regular graph
A graph is regular if every vertex is of the same degree
It is k-regular if every vertex is of degree k
Is it possible to have a 4-regular graph with 10 edges ?
From theorem 11.2,
Two nonisomorphic 4-regular graphs

9. Planar Graphs Definition
A graph G is called planar if G can be drawn in the plane with its edges intersecting only at vertices of G.
Such a drawing of G is called an embedding of G in the plane.
That is,
planar is the property of a graph that it can be drawn in the plane without crossings
A graph embedding (or imbedding) is a drawing with no crossings at all.

10. Examples K1 K2 K3 K4
embedding
Planar Graphs K5 K5
Non-planar Graph K5

11. Examples
Planar Graphs

12. 각각 서로 다른 vertex set 간에만 edge가 존재함
Bipartite Graphs
Definition
A graph G = (V,E) is called bipartite, if V = V1  V2 with V1  V2 =  , and every edge of G is of the form {a, b} with a  V1 and b  V2.
각각 서로 다른 vertex set 간에만 edge가 존재함
Isomorphic 10 11 00 01 11 10 01 00 V1 V2

13. An Example
Bipartite Graphs

14. Bipartite Graphs That is,
A graph is bipartite if its vertices can be partitioned into two subsets (the parts, or partite sets) so that no two vertices in the same part are adjacent.
A graph whose vertices can be divided into two disjoint sets U and V such that every edge connects a vertex in U to one in V; that is, U and V are independent sets
[wikipedia]

15. An Example Q3 Hypercube 100 101 110 111 011 010 001 000 Isomorphic 100V1 V2

16. Complete Bipartite Graphs
Km,n
If each vertex in V1 is joined with every vertex in V2, we have a complete bipartite graph.
If |V1| = m, |V2| = n, the graph is denoted by Km,n.
That is,
complete bipartite graph is a bipartite graph Km,n whose vertex set has two parts of sizes m and n, respectively, such that every vertex in one part is adjacent to every vertex in the other part.
9 edges
K3,3
K2,3

17. Non-planar Graph K3,3
Isomorphic
K3,3

18. Subdivision A graph G = (V,E) be a loop-free undirected graph.
An elementary subdivision of G results when an edge e = {u, w} is removed from G and then the edges {u, v}, {v, w} are added to G-e where v V. e d u w c e d u w c v e d u w c v x
G-e: unchanged vertex set, spanning tree

19. Subdivision
That is,
subdivision (of an edge) is the operation of inserting a new vertex into the interior of the edge, thereby splitting it into two edges
For example, the edge e, with endpoints {u,v} can be subdivided into two edges, e1 and e2, connecting to a new vertex w

20. These are homeomorphic
Homeomorpic
The loop-free undirected graphs G1 = (V1, E1) and G2 = (V2, E2) are called homeomorphic, if
(a) they are isomorphic or
(b) they can both be obtained from the same loop-free undirected graph H by a sequence of elementary subdivisions. e d a b c v
These are homeomorphic e d a b c e d a b c y w z
By elementary subdivision H

21. G and H are homeomorphic
Example
In other words, G and H are homeomorphic if there is an isomorphic from some subdivision of G (that is, G’) to some subdivision of H (that is, H’)
G and H are homeomorphic H G
By elementary subdivision
By elementary subdivision
G’, H’
are isomorphic
If G' is the graph created by subdivision of the outer edges of G and H' is the graph created by subdivision of the inner edge of H, then G' and H' have a similar graph drawing.
herefore, there exists an isomorphism between G' and H', meaning G and H are homeomorphic.

22. G and H are homeomorphic
Example
G and H are homeomorphic H G
By elementary subdivision
By elementary subdivision
H and H’ are homeomorphic
G’ and H’ are isomorphic
G and G’ are homeomorphic H’ G’
In Greek:
homeo (homoios): “similar”
Morphe “shape”

23. Example G1 is obtained from G by means of one elementary subdivision
G2 is obtained from G by two elementary subdivisions
G1 and G2 are homeomorphic to G
G3 is obtained form G by four elementary subdivisions. Also, G3 can be obtained from G1 & G2. (G3 is homeomorphic to G, G1 & G2)

24. Example
Homeomorphic graphs may be isomorphic except, possibly, for vertices of degree 2.
In particular, if two graphs are homeomorphic, they are either both planar or both nonplanar. K5 e d a b c v e d a b c y z w

25. Theorem 11.5 : Kuratowski’s Thm.
A graph is nonplanar if and only if it contains a subgraph that is homeomorphic to either K5 or K3,3. a e d a b j f c g h i f e j g b
Isomorphic i h d c
(b) Subgraph of the Petersen graph that is homeomorphic to K3,3)
(a) Petersen graph
* Subgraph
If G=(V,E) is a graph, then G1=(V1, E1) is called a subgraph of G if   V1  V and E1  E where each edge in E1 is incident with vertices in V1.
cf) spanning subgraph
* Homeomorphic :

26. Theorem 11.5 : Kuratowski’s Thm.d a b j f c g h i
K3,3 에 homeomorphic한 graph G’ e d a b j f c g h i G’ G
붉은색 edge를 지우면 G’이 됨.
즉, G는 G’을 subgraph로 가짐 a e d b c j g h f i
Isomorphic
Petersen graph

27. Regions Edge에 의하여 둘러싸인 영역 Embedded planar graph에서 정의
Any of the pieces of surface subtended by the graph (그래프에의해 범위가 정해진 영역)
A region of an imbedded graph is, informally, a piece of what results when the surface is cut open along all the edges.
From a formal topological viewpoint, it is a maximal subsurface containing no vertex and no part of any edge of the graph.
Infinite
region R1 R2 R3 R4 R1 R2 R3 R4

28. Regions Degree of region, deg(R)
the number of edges in a (shortest) closed walk about the boundary of R
Infinite
region R1 R2 R3 R4 R1 R2 R3 R4

29. Examples Even for a graph with loops and a multigraph R1 R2 R3 R1 R2

30. Theorem 11.6 Euler’s formula (v – e + r = 2)
Let G = (V,E) be a connected planar graph or multigraph with |V| = v and |E| = e.
Let r be the number of regions in the plane determined by a planar embedding of G.
Then, v – e + r = 2. R1 R2 R3 R4
Euler’s formula
; v – e + r = 2
v = 5
e = 7
r = 4

31. Examples Even for a graph with loops and a multigraph R1 R2 R3 R1 R2
v – e + r = 2
v = 5
e = 6
r = 3
v – e + r = 2
v = 4
e = 8
r = 6

32. Examples v – e + r = 2 v = 1 e = 0 r = 1 v – e + r = 2 v = 1 e = 1

33. Corollary 11.3 Two Inequalities
Let G = (V,E) be a loop-free connected planar graph with |V| = v and |E| = e > 2, and r regions. Then, 3r  2e and e  3v-6. (only )
Proof
Since each region contains at least three edges, Degree of each region  3
3r   deg(Ri) = 2e  3r  2e
2 = v – e + r  v – e + (2/3) e = v – (1/3) e
 e  3v-6 r1 r2
3r <= 2e 관계식:
먼저 SUM(deg(Ri)) = 2e라는 것을 연상함. 즉, 각 region Ri의 degree를 계산함
각 region의 degree는 가정 조건에서 edge갯수가 3개이상이고 loop가 없다고 했으므로
각 region의 degree는 3개 edge 이상일것임. 즉, 여기서의 총 edge 수는 region의 개수 : r* 3 이상이될것임
(3) 그래서 3* r 이상이라고 했음 3*r <= SUM(deg(Ri)).
그리고 region degree 수식 사용
최소 3개 이상의 edge로 만들어지는 region. 이때 당연히 degree는(각 region의 edge 개수)는 region * 3이 됨. 3r

34. Examples By loop free Inequality holds. It is multigraph !!
Let G = (V,E) be a loop-free connected planar graph with |V| = v and |E| = e > 2, and r regions. Then, 3r  2e and e  3v-6.
By loop free
Inequality holds.
It is multigraph !!
3r  2e , r=2, e=3
e  3v-6, r=3, v=3

35. Examples Inequality holds. K4 v = 4 e = 6  6 = 3v – 6 r = 4 e = 6

36. Is it planar ?
Corollary 11.3 shows just a necessary condition for testing planarity of graphs
3r  2e
e  3v-6
Planar graph
Contrapositive
Nonplanar Test
We know that
How to test planarity ?
To prove "If P, Then Q" by the method of contrapositive means to prove "If Not Q, Then Not P".
Method of Contradiction: Assume P and Not Q and prove some sort of contradiction.
Method of Contrapositive: Assume Not Q and prove Not P.

37. Nonplanar Test (e > 3v-6)
3r  2e
e  3v-6
K3,3
v = 6
e = 9
< 12 = 3v – 6
Not so good !
v = 10
e = 15
< 24 = 3v – 6
v = 5
e = 10
> 9 = 3v – 6
Petersen graph K5

38. Planar or Nonplanar ? Is it planar or nonplanar ? Kuratowski’s theorem
v = 5
e = 6
< 9 = 3v – 6
Kuratowski’s theorem
A graph is nonplanar if and only if it contains a subgraph that is homeomorphic to either K5 or K3,3.

39. Planarity Testing
There are many graphs which satisfy the inequalities but are not planar
Therefore, we need other way to decide the planarity
Observation 1
Not all graphs are planar.
For example, we know K5 and K3,3 are not planar.

40. Planarity Testing Observation 2
If G is a planar graph, then every subgraph of G is planar;
We usually stated observation 2 as follows
Observation 2a
If G contains a nonplanar graph as a subgraph, then G is nonplanar. For example, following graph is nonplanar.
Since it contains K5 and K3,3 as a subgraph

41. Planarity Testing Observation 3
If G is a planar graph, then every subdivsion  of G is planar, we usually stated observation 3 in the following way.
Observation 3a
If G is a subdivision of a non-planar graph, then G is non- planar.
For example, following graph is non-planar,
Since it contains K5 and K3,3 as a subgraph

42. More on Nonplanarity & Petersen Graph
A graph is nonplanar if and only if it contains a subgraph that is homeomorphic to either K5 or K3,3. e d a b c j g h f i e d a b j f c g h i
(b) Subgraph of the Petersen graph that is homeomorphic to K3,3)
(a) Petersen graph
Subgraph
If G=(V,E) is a graph, then G1=(V1, E1) is called a subgraph of G if   V1  V and E1  E where each edge in E1 is incident with vertices in V1
Homeomorphic
isomorphic or can be obtained from elementary subdivisions.

43. More on Nonplanarity & Petersen Graph
Petersen graph is nonplanar. Any nonplanar graph has as minors either K5 or K3,3.
Petersen graph has K5 and K3,3 as minors.
Minor: graph H is called a minor of the graph G if H is isomorphic to a graph that can be obtained by zero or more edge deletions or edge contractions on a subgraph of G.
Edge contraction is the process of removing an edge and combining its two endpoints into a single node. e d a b c j g h f i
Ex of Minor) G H
Petersen graph
Graph H is a minor of graph G
Construct a subgraph of G by eliminating the dashed-line
contract the gray edge (combining the two vertices it connects)

44. More on Nonplanarity & Petersen Graph
Petersen graph has K5 and K3,3 as minors. e d a b c j g h f i a
Edge
contractions f f e j g b j g i h i h d c
Petersen graph K5

45. More on Nonplanarity & Petersen Graph
Petersen graph has K5 and K3,3 as minors. c e i b d h e d a b c j g h f i e d a b c j g h f i
Edge deletions
Isomorphic to K3,3 e b i h d c
Petersen graph
Edge
contractions e d a b c j g h f i
Homeomorphic to K3,3 ! (by subdivision!) e d a b c j g h f i
subgraph