Planar graph( Algorithm and Application )

Planar graph( Algorithm and Application )

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  Planar Graph Algorithm andApplications
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  Hello! I am AbdullaAl Moin Computer Science & Engineering Daffodil International University 2
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  Submitted To! Rubel Sheikh Lecturer ComputerScience & Engineering Daffodil International University 3 Under Course : Course – CSE 236 Math For Computer Science
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  “ PLANAR GRAPH 4
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  History 5 Graph
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  Instructions What is PlanarGraph? 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. 6
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  Some Example OnPlaner Graph 7
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  Theorems Kuratowski’s theorem Plane : Agraph 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! 8
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  Euler’s Theorem Theorem (Euler) Let, Gbe 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 9
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  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. 10
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  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. 11
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  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) = ∅ . 12
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  Applications. Applications : – Visuallyrepresenting 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. 13
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  Thanks! Any questions? 14