Analysis of Connectivity in Graphs

1
Analysis of Connectivity in Graphs

• Harini Krishnamurthy
• Karthik Nandakumar
• Kuo-Liang Chang

2
Goals

• Obtain a histogram of vertex-connectivity for all
  graphs of order lt 10
• Compute minimum number of edges needed to have a
  k-connected n-order graph
• Analyze the percentage of connected graphs that
  satisfy the equality in Whitneys theorem
• i.e. ? ? ?

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Finding Connectivity of a graph
Initialize ? n Form the network N G (V, E,
u, v).
Get adjacency matrix from nauty
Find Max-Flow in N, f(N)
if f(N) lt ? ? f(N)
4
Transformation of Graphfrom graph to the
corresponding network

x
x-
y-
s-
s
t
t-
Source
Sink
y
5
Max-Flow Algorithm

• An Experimental Comparison of Min-cut / Max-flow
  Algorithms for Energy Minimization in Vision
• - Yuri Boykov and Vladimir Kolmogorov
• Similar to Dinics algorithm
• Builds search trees to find the augmenting paths
  but reuses this tree for further iterations
• Drawback does not find the shortest augmenting
  path
• Theoretically, time complexity of this algorithm
  is O(mn2) but authors have experimentally shown
  that on typical problem instances in vision, it
  is much faster compared to other standard
  algorithms.

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From Max-Flow to Vertex Connectivity

• If G is a complete graph, ? n-1 and stop
• If G is disconnected, ? 0 and stop
• ? n
• i 0
• Step to determine if connectivity has been
  computed
• if i ? ?, then i i1, continue. Else
  stop.
• 6. j i1
• 7. Update ? until ?(G) is found
• 7.1 if j n1, return to step 5
• 7.2 if uv E(G), construct network N
  (V, E, u, v-)
• 7.3 Determine f(N).
• 7.4 if f(N) lt ?, ? f(N), else
  continue
• 7.5 j j1. Return to step 7.1

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Histogram of connectivity (1)
disconnected
1 connected
2 connected
3 connected
4 connected
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Histogram of connectivity (2)
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Equality in Whitneys theorem? ? ?
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Observations

• As the order n increases, the percentage of
  graphs having lower connectivity decreases.
• The minimum number of edges needed to have a
  k-connected n-order graph is
• n ?(k-2)n/2?
• (If k 1, then it is n-1)
• Of all connected graphs of order, n lt 10,
• nearly 96 satisfy the equality ? ? ? and
• nearly 97.5 of the graphs have ? ?