Introduction to Planarity Test

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Introduction to Planarity Test

• W. L. Hsu

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Plane Graph

• A plane graph is a graph drawn in the plane in
  such a way that no two edges intersect
• Except at a vertex to which they are both
  incident
• A planar graph is one which is isomorphic to a
  plane graph
• Namely, it has a plane embedding

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Planar Graphs
4
Planar Graph EmbeddingClockwise edge ordering
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Issues in Planarity Test

• If you can find a planar embedding, then the
  graph is planar.
• How do you determine if a graph is not planar?
• This is the more difficult part of many
  recognition algorithm, namely, deciding when a
  graph does not belong to a class
• Get a certificate for non-planar graphs
• Or alternatively, you have tried all possible
  ways but still fail to embed the graph in the
  plane (proof by exhaustion)
• Use counting argument

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Basic Non-Planar Graphs
K3,3
K5
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Eulers Theorem (1752)

• Eulers theorem
• Let G be a connected plane graph, and let f be
  the of faces of G.
• Then n f m 2
• Prove by induction on the of edges.
• Corollary. m ? 3n 6
• First show that 3f ? 2m since every face is
  bounded by at lest 3 edges

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K5 and K3,3 are non-planar

• If K5 is planar, then by previous Corollary, we
  have 10 ? 9. ??
• K3,3 is bipartite. Assume it is planar, then
  every face is even (has at least 4 edges).
• Hence 4f ? 2m or 2f ? m .
• Do not adopt the previous Corollary directly
• Namely, 10 2f ? m 9. ??

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Kuratowskis Theorem

• Two graphs are homeomorphic if they can be
  obtained from the same graph by inserting new
  vertices of degree 2 into its edges
• A graph is planar if and only if it contains no
  subgraph homeomorphic to K5 or K3,3
• The latter are referred to as Kuratowski subgraphs

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Planarity Test
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How do you draw a planar graph without regret ?

• This means that, besides keeping the current
  embedding planar, your embedding can also keep
  future options open.
• You will have to design an embedding scheme
  rather than obtain a physical (???) embedding

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Prior Results

• 1st approach
• Hopcroft and Tarjan 1974,first O(m) time.
• PATH ADDITION
• 2nd approach
• Lempel, Even and Cederbaum1967, O(n2) time
• VERTEX ADDITION
• st-numbering, consecutive ones testing
• Booth and Lueker 1976 used PQ-trees to test the
• consecutive ones property in O(mn) time
• 3rd approach
• Shih and Hsu 1999 used PC-trees for recognition
  and embedding.
• EDGE ADDITION

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A Brief Intro. to the Vertex Addition Approach of
LEC
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Vertex Addition Approach of LEC

1. Keep the current partial planar graph connected
2. Keep those non-added vertices a connected
   subgraph (i.e. in the same face).
3. Apply a consecutive ones test every time a new
   vertex is added

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st-numbering (I)

• Consider a 2-connected graph G. Pick any two
  adjacent vertices s and t.
• Order the vertices of G into s, v(1), ..., v(k),
  t such that

s
t
v(i)
v(i1)
t
s
v(i)
v(i1)
v(i1),, t must be imbedded in the same face
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St-numbering(II)
t
s
v(i)
v(i1)
t
s
v(i)
v(i1)
Depth-First-Search
t
s
v(i)
v(i1)
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Bush Form (1)
1s
6t
2
3
5
6
2
3
5
(a) B1
(a)
1
2
6
3
5
4
5
3
6
3
5
4
5
3
(b) B2
(b)
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Bush Form (2)
1
2
6
3
5
4
5
3
5
4
5
3
3
6
(c)
(c) B2
1
2
3
6
5
4
5
6
5
4
5
6
4
4
6
(d) B3
(d)
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Bush Form (3)
1
2
3
6
5
6
5
4
5
6
4
5
4
4
6
(e) B3
(e)
1
2
3
4
5
4
5
6
4
6
5
5
6
6
5
6
(f)
(f) B4
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Bush Form (4)
1
3
2
4
6
5
5
6
6
5
6
5
5
6
6
5
(g) B4
(g)
1
2
3
4
5
6
6
6
6
6
6
6
6
(h)
(h) B5
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Bush Form (5)
1
2
3
4
5
6
(i) GG6B6
(i)