A graph is said to be planar if it can be drawn in a plane so that no edge cross. Example: The graph shown in fig is a planar graph. Region of a Graph: Consider a planar graph G=(V, E). A region is defined to be an area of the plane that is bounded by edges and cannot be further subdivided.

1. Planar Graph
Algorithm and Applications

2. Hello!
I am Abdulla Al Moin
Computer Science & Engineering
Daffodil International University
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3. Submitted To!
Rubel Sheikh
Lecturer
Computer Science &
Engineering
Daffodil International University
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Under Course :
Course – CSE 236
Math For Computer
Science

4. “
PLANAR GRAPH
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5. History
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Graph

6. Instructions
What is Planar Graph?
Plane :
Our world has three dimensions, but there are
only two dimensions on a plane :
Length(x) and Width(y) makes a Plane. No
thickness and goes on forever.
Planar Graph :
It is a simple Graph. Can be drawn on the
plane without edge crossings.
Details®
In graph theory, a planar graph is a graph that can
be embedded in the plane, i.e., it can be drawn on
the plane in such a way that its edges intersect
only at their endpoints. In other words, it can be
drawn in such a way that no edges cross each
other.
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7. Some Example On Planer Graph
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8. Theorems
Kuratowski’s theorem
Plane :
A graph is planar, if and only if it does not
contain the K5 and the K3,3 as a
homeomorphic subgraph / as a minor.
H is a minor of G, if H can be obtained from G by a
series of or more deletions of vertices, deletions of
edges, and contraction of edges.
Does not yield fast recognition algorithm!
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9. Euler’s Theorem
Theorem (Euler)
Let,
G be a connected plane graph with n vertices,
m edges, and f faces.
Then n + f – m = 2.
Proof. By induction.
– True if m=0.
– If G has a circuit, then delete an edge
– If G has a vertex v of degree 1, then delete
v
Euler’s Theorem Corollaries.
Corollary 1: Let G be a connected planar simple
graph
with e edges and v vertices. Let r be the
number of regions in a planar
representation of G.
Then r = e - v + 2.
# of edges, e = 6
# of vertices, v = 4
# of regions, r = e - v + 2 = 4
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10. Euler’s Theorem Corollaries. (Cont..)
Corollary 1:
For any simple, connected, planar
– graph G, with |E| > 2, the following holds:
– |E| ≤ 3n – 6
Proof: Each face is bounded by at least 3
edges, so:
Σd( fi) ≥ 3f
Substitute 3f with 6 – 3n + 3|E|, and use the
lemma.
Corollary 2:
For any simple connected bipartite planar graph G,
with |E| > 2, the following holds:
|E| ≤ 2n – 4
Proof: Each face of G is bounded by at least 4
edges. The result then follows as for the previous
corollary.
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11. Euler’s Theorem Corollaries. (Cont..)
Corollary 3:
In a simple, connected, planar graph there exists at least one vertex of degree at most 5.
Proof:
Without loss of generality we can assume the graph to be connected, and to have at least three
vertices.
If each vertex has degree at least 6, then we have
6n ≤ 2m, and so 3n ≤ m.
It follows immediately from previous Corollary that
3n ≤ 3n — 6,
which is a contradiction.
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12. Planarity Testing
Simple tests :
Following the simplifications,
two elementary tests can be applied:
– If e < 9 or n < 5
then the graph must be planar.
– If e > 3n – 6
then the graph must be non-planar.
If these tests fail to resolve the question of
planarity, then we need to use a more
elaborate test.
Planarity Testing Algorithms:
Notations: Let B be any bridge of G relative to H.
B can be drawn in a face f of H', if all the points of
contact of B are in the boundary of f.
F(B,H): Set of faces of H' in which B is draw able.
The algorithm finds a sequence of graphs G1, G2,
…, such that
Gi ⊂ Gi+1.
If G is non-planar then the algorithm stops with the
discovery of some bridge B, for which
F(B,Gi) = ∅ .
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13. Applications.
Applications :
– Visually representing a network (e.g., social
network, organization structure, data bases
(ER-diagrams), software (e.g., UML-
diagrams), flow charts, phylogenetic trees
(biology, evolution), …
– Design of “chip” layout (VLSI)
Applications : (Computer Vision)
In ComputerVision there is an abundance of
problems that can be addressed as finding a
minimal cut through a planar graph.
such as shape matching, image segmentation or
cyclic time series.
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14. Thanks!
Any questions?
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