MC 302 Graph Theory Thursday, 102804

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MC 302 Graph TheoryThursday, 10/28/04

• Today (Questions?)
• Go over rest of Worksheet 7
• k-connected and k-edge-connected graphs
• Whitneys Theorem
• Mengers Theorem
• Application k-connected networks
• Todays reading exercises
• Reading still 4.3, Exercise 13
• Also read Bondy Murty Section 3.3, Exercise
  3.3.1
• Link from class web site, or
• Direct link http//www.ecp6.jussieu.fr/pageperso/
  bondy/books/gtwa/gtwa.html
• Reminders
• HW 4 due Monday at 5 PM
• Dont forget to VOTE on Tuesday!

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k-Connected and k-edge-Connected Graphs

• For a connected graph G, we now have that ?(G) is
  the minimum of vertices whose removal
  disconnects or trivializes G, and likewise ?(G)
  for edges.
• We say a graph G is k-connected if ?(G) ? k.
• We say a graph G is k-edge-connected if ?(G) ?
  k.
• Since ?(G) ? ?(G), we have if ?(G) ? k, then also
  ?(G) ? k. So a k-connected graph is always
  k-edge-connected, but not necessarily vice versa.
• We will focus primarily of vertex-connectivity.
• Which are 1-connected, 2-connected, 3-connected
• Trees, cycles, Q2, Q3, Kn, Kp,q

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k-Connected Graphs

• Alternative Definition A (simple) graph is
  k-connected if at least k vertices must be
  deleted to disconnect or trivialize it.
• A 1-connected graph is the same as a connected
  graph.
• A 2-connected graph is a connected graph with no
  cut vertices.
• Note that a 2-connected graph is also
  1-connected.
• In general if k gt m and G is k-connected, then
  its also m-connected (but not necessarily vice
  versa).
• The highest k for which a graph G is k-connected
  is ?(G).
• The higher k is, the more reliable a network
  based on this graph will be.

4
Generalizing the Unique Path Property Whitneys
Theorem

• We know A graph with 2 or more vertices is
  1-connected IFF there is at least one path
  between each pair of distinct vertices.
• Whitneys Theorem A graph with 3 or more
  vertices is 2-connected IFF there are at least
  two internally disjoint paths between each pair
  of vertices.

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More on Whitneys Theorem

• Corollary A graph G is 2-connected IFF every
  pair of vertices, u, v, are contained in a cycle.
• Verify above Corollary and Theorem for graphs on
  previous slides.
• Generalized Whitneys Theorem A simple graph is
  k-connected IFF there are at least k internally
  vertex-disjoint paths between each pair of
  vertices.

6
Mengers Theorem

• A result related to Whitneys Theorem
• Mengers Theorem. Let u and v be distinct
  nonadjacent vertices in G. Then the maximum
  number of internally disjoint paths from u to v
  equals the minimum number of vertices in a set
  that separates u from v.
• This is also related to the Max-Flow Min-Cut
  Theorem in Chap. 10 which is about maximimizing
  flow in a directed network with a source and
  sink

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Application K-connected networks (BM Section
3.3)

• Reliable networks need to be connected, but
  k-connected for k gt 1 is even better.
• For every n, there is a graph on n vertices that
  is 1-connected what is it?
• For every n, is there a graph on n vertices that
  is 2-connected?
• What about k-connected?

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k-Connected graphs on n vertices.

• Since ?(G) ? ?(G), it follows that a k-connected
  graph on n vertices has at least edges.
• Can we always find a graph G with n vertices that
  is k-connected?
• YES result of Harary, 1962. There is a such a
  graph, named Hk,n

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Constructing Hk,n for k even

• Start with a cycle on n vertices
• Add extra edges to make it k-connected.
• Case 1. k even, k 2r. (See BM for other
  cases).
• Start with n vertices, 0, 1 , , n-1.
• For each vertex i, add edges to r preceding and r
  following vertices.
• Then

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Theorem for k even

• Theorem. For k 2r even, Hk,n is k-connected.
• Proof
• Let V be any set of vertices with V lt 2r, and
  let i and j be distinct vertices in G V .
• Look at the two sections S1 i, i1, , j, and
  S2 j, j1, , i.
• At least one of these sets, say S1, contains
  fewer than r vertices of V
• This implies there is a path in S1 from i to j
• Therefore V does not disconnect Hk,n.