CSCI 4260 MATH 4150

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CSCI 4260MATH 4150

• GRAPH THEORY

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A note on the quiz

• There were some excellent solutions as well as
  some really clueless ones.
• Think of the quiz as a warning if you are having
  too much difficulty, you may have a hard time in
  this course.
• Solution review the proofs and ideas after every
  lecture.
• More quizzes will be coming.
• Special thanks to some of you for the really
  thorough grading!

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Todays topic

• Trees nicest graphs
• Lets start with an example

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Telephone network
What is nice about this structure?
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Definition

• Acyclic graph no cycles
• Tree acyclic and connected
• Rooted trees Child parent relationship

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Theorem

• A graph is a tree iff every two vertices of G are
  connected by a unique path.

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Leaves

• Definition leaves (or end vertices)
• TheoremEvery nontrivial tree has at least 2
  leaves.

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Trees have an inductive structure
Smaller trees
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Theorem

• Every tree of order n has size n-1

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Forests

• A forest is acyclic but not necessarily connected
• Collection of a bunch of trees

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Theorem

• Every forest of order n with k components has
  size n-k

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Theorem

• The size of every connected graph of order n is
  at least n-1
• Try for n 1, 2, 3
• Suppose the theorem is false. Then there exists a
  graph with at most n-2 edges.
• Let G be the smallest sized such graph and n be
  its order. Clearly ngt3.
• G has a leaf proof
• Can you find a smaller sized connected graph of
  size less than n-1?

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Theorem

• If G has any two properties below, then its a
  tree
• Connected
• Acyclic
• m n - 1

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Weighted Graphs

• So far, G (V,E)
• We can also have weights on edges (possibly on
  nodes too)
• For example
• Nodes cities
• Two cities are connected by an edge if there is a
  road connecting them
• The weight of an edge is the length of the road.

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Spanning subgraphs

• Definition?
• Since trees are nice, it is common to look for
  spanning trees.
• This raises the minimum spanning tree problem
  (MST)
• Find the spanning tree whose weight (sum of the
  weights of its edges) is minimized.

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Kruskals Algorithm

• Input Graph G Output Spanning Tree T
• Let e1, e2, ., em be the edges sorted in
  increasing weight
• T ??, i 1
• While T is not a spanning tree
• If T ei has cycles
• Throw ei away
• Else
• T ? T ei

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Theorem

• Kruskals algorithm produces an MST.
• Proof suppose not
• T output of Kruskal
• Let H be an MST (Among all MSTs, pick the one
  that has a max number of edges in common with T)
• We have w(H) lt w(T)
• Suppose e1, e2, ., ei-1 are common

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Proof (cont)

• Let G0 ? H ei
• The graph G0 has a cycle C. Let e0 be an edge
  on C, that is not on T (why does e0 exist?)
• Let T0 ? G0 - e0
• w(T0) w(H) w(ei) w(e0)
• Since H is an MST w(e0) w(ei)
• If i 1, w(ei) w(e0)

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Proof (cont)

• If i 1, w(ei) w(e0)
• If i gt1
• ei the edge added at iteration i (min weight at
  the time)
• But we could have added e0 as well. How can this
  be?
• In all cases T0 has more edges in common with T
  than H. A contradiction!