CSCI 4260MATH 4150

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CSCI 4260-MATH 4150

• Lecture 11

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Directed Graphs

• So far we have been mostly concerned with
  undirected graphs
• Today, we will focus on directed graphs

(u,v) is an arc the order is important u is
adjacent to v v is adjacent from u
v
u
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Indegree vs. Outdegree

• Indegree of v id(v) u u is adjacent to
  v
• Outdegree of v
• od(v) u u is adjacent from v

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Theorem
For any directed graph G (V,E)
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Orienting an undirected graph, underlying graph
G is the underlying graph of H
H is an orientation of G
Note that H does not have the arcs (u,v) and
(v,u) simultaneously
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Connectivity in directed graphs

• Our paths are now directed.
• Weakly connected the underlying (undirected)
  graph is connected
• Strongly connected there is a directed u-v path
  AND a directed v-u path for all u,v
• In this case, we say the graph is strong.

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Some basic properties

• In the next few theorems, we will see the
  directed version some earlier ones.

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(Warm-up) Theorem

• If a digraph D contains a u-v walk of length l,
  then D contains a u-v path of length at most l

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Theorem

• A digraph is strong iff it contains a closed
  spanning walk

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Theorem

• A non-trivial, connected graph D is Eulerian iff
  od(v) id(v) for every vertex v.

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Eulerian ? od(v) id(v) ?v

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od(v) id(v) ?v ? Eulerian

• Pick an initial vertex u and find the longest
  trail T
• T must be closed, i.e a circuit.
• Suppose T does not contain all edges
• Let D D T
• There must be a vertex w on T that is incident to
  an edge not in T
• Note that od(v) id(v) for all v in D
• Apply the same argument, there must be a circuit
  starting from w
• But then T can be made longer a contradiction!

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Theorem

• A non-trivial (undirected) graph G has a strong
  orientation iff G contains no bridge.

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Strong orientation ?no bridge

• Take the contrapositive
• If there is bridge,

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No bridge ? Strong orientation

• Remember, no bridge means that every edge is on a
  cycle
• The main idea is to cover the graph with cycle
  and to choose an orientation for each cycle
• Lets prove this formally

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Tournaments

• A tournament is an orientation of a complete
  graph
• Think of it as a real tournament
• Vertices teams
• All teams play with each other
• u ? v means u won the game
• A transitive tournament is defined as arc (u,v)
  and (v,w) ? arc (u,w)

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Theorem

• A tournament is transitive iff it has no cycles

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Transitive ? no cycles

• Suppose a transitive tournament has a cycle

vk
v1
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No cycles ? Transitive

• If (u,v) and (v,w) are arcs in T
• (w,u) can not be an arc in T
• This means (u,w) is an arc

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Theorem

• If u is a vertex of maximum outdegree in a
  tournament T, then d(u,v) ? 2 for every vertex

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Proof
Max outdegree vertex, degree k
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Interpretation

• If A is the team that won most matches,
• For any team B
• Either A defeated B or
• A defeated a team that defeated B

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Theorem

• Every tournament contains a hamiltonian path

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Proof

• Suppose not. Take the longest path. There must be
  some vertex not on it

Cant have these red edges
v
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Proof
v
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Interpretation

• In every tournament, we can find a sequence of
  teams a,b,c,d, where a beat b, and b beat c, and
  so on

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But who is the best?
Remember transitive ? no cycles
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Theorem

• Every vertex in a non-trivial strong tournament
  belongs to a triangle

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W
U

• U ??? and W ???Why?
• There must be an arc from w to some vertex in U

w
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Theorem

• A nontrivial tournament T is strong iff T is
  Hamiltonian

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Hamiltonian ? Strong

• ?

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Strong ? Hamiltonian

• Let C be the maximum length cycle and for
  contradiction assume that there exists a vertex
  v that is not on C

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Case I
w
u
Cant happen
v
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Case II ?
v
u
So, its the same case
v
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Case II
It must be that v is either adjacent to or
adjacent from all vertices on C.
v
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Cycle C
U ??
W ??
Why must this edge exist?
Can you find a bigger cycle?
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Recap

• Transitive tournaments have an absolute winner
• Strong tournaments have hamiltonian cycles
• Hence are not transitive

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Also

• Read sections 7.3 and 7.4 in the book
• Its fun
• Project proposal deadline is coming soon!
• Next Thursday there will be an overview lecture,
  given by your TA
• Come prepared

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Next few weeks
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HW2 is out today

• You can discuss the problems with a single
  classmate
• You have to put her/his name on your solution
• You have to write your solutions independently
• The school is enforcing a very strict cheating
  policy.
• Basically, any type of cheating is reported all
  the way up to the dean
• So, please dont!