Graph Theory

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Graph Theory

• ???? ????? ???
• 2011.9

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Textbook

• G. Agnarsson and R. Greenlaw, Graph Theory
  Modeling, Applications, and Algorithms, Pearson,
  2007.
• G. Chartrand and O. R. Oellermann, Applied and
  Algorithmic Graph Theory, McGraw-Hill, 1993.

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Contents

• Ch1 - Introduction to Graph Theory
• Ch2 - Basic Concepts in Graph Theory
• Ch3 - Trees and Forests
• Ch4 - Spanning Trees
• Ch5 - Fundamental Properties of Graphs and
  Digraphs
• Ch6 - Connectivity and Flow
• Ch7 - Planar Graphs
• Ch8 - Graph Coloring
• Ch10 - Independence, Dominance, and
  Matchings
• Ch12 - Graph Algorithms

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Chapter 1 Introduction to Graph Theory
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1.2 Why study graphs?

• Problem 1.1 The Bridges of Konigsberg

Problem Make a round trip through downtown
Konigsberg, traversing each bridge exactly once.
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B1
I1
I2
B2
Q ????????,?????????,??????

• Ans ?????????,???????????,????????,??????????????
  ???
• ? ????????????????
• ? ??????? (Chapter 5)

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• Problem 1.2 World Wide Web Communities

Complete Bipartite Graph
??????
?????
????
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• Problem 1.3 Job Assignments

Jobs
qualified
Applicants
Bipartite Graph
Problem 1.8 Is the company able to meet its
hiring need? If so, provide a possible set of
hires that meet their needs.
Ans No
(Ch10 Matching)
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• Problem 1.4 Storing Volatile Chemicals

Problem C1, C2, , C7 ??????????,?????????, (An
edge between Ci and Cj indicates a grave danger
in storing these chemicals in the same
warehouse.) ????????
1
2
4
3
3
Ans 4
1
(Ch8 Graph Coloring)
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1.3 Mathematical Preliminaries

• Set, element, empty set, subset,
• union, intersection, disjoint, difference
  (A\B),
• cardinality (A, ?A???????)
• symmetric difference of A and BA?B (A\B) ?
  (B\A)
• power set of S P(S) all subsets of S
• k-tuple (a1, a2, , ak)
• Cartesian product of A1, A2, , Ak is A1?A2??Ak
  (a1, a2, , ak) ai ? Ai for each i

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1.4 The Definition of a Graph

• A graph or a general graph is an ordered triple
  G (V, E, ?), where1. V ? ?.2. V?E ?.3. ?
  E ? P(V) is a map such that ?(e) ? 1, 2
  for each e?E.
• Vertex (?) element of V (V????V(G))
• Edge (?) element of E (E????E(G))
• ? edgemap
• ?(e) endvertices (???) of the edge e
• (Note V and E can be
  infinite.)

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Example
G(V, E, ?) Vu1, u2, u3, u4, u5 Ee1, e2, e3,
e4, e5, e6
?(e1)u1, u2 ?(e2)?(e3)u1, u3 ?(e4)u2,
u3 ?(e5)u3, u4 ?(e6)u4
u5 is called isolated.
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• u, v vertices of a graph G
• u is called an endvertex (??) of e.
• u and v are adjacent (or neighbors)
• u and e are incident. (adjacent???????,???????,
  incident???????)
• loop

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1.5 Examples of Common Graphs

• Simple graph a graph having no multiple edges or
  any loop. ? can be omitted ? G(V, E)

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Exercise 11, 12
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1.6 Degrees and Regular Graphs
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f u or
hbors
is the
Exercise 17
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pf. ???degree???,?????????,
??degree???????????
degree sum 12E(G) 6
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(degree?????,???????)
pf. If the number of vertices with odd degree is
odd, then the degree sum must be odd. ??
The null graph Nn is 0-regular. The cycle Cn is
2-regular. The complete graph Kn is
(n-1)-regular. The complete (m,n)-bipartite graph
Km,n is a regular graph if and only if
mn. Every k-regular graph on n vertices has kn/2
edges.
Exercise 13
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1.7 Subgraphs
is a subgraph of G if
We write G ?G.
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If Ww1, w2, , wm, we write Gw1, w2, , wm
instead of Gw1, w2, , wm.
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Exercise 14, 15, 19
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1.8 The Definition of a Directed Graph
(h??eta)
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G (V, E, h)
V u1, u2, u3, u4, u5
E e1, e2, e3, e4, e5, e6
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(???????????,??????)
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Simple digraph a digraph without directed loops
and parallel directed edges.
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1.9 Indegrees and Outdegrees in a Digraph
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and the
N -(u3) u1, u2
N (u3) u1, u4
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( ???indegree?? outdegree?? ?? )
A directed cycle is balanced and regular.
Exercise ??Draw a nonregular balanced
digraph of 5 vertices. 26