1 / 39

Planar Graphs

Great Theoretical Ideas In Computer Science. Planar Graphs. Lecture 22. CS 15-251. The clique on 4 nodes. A graph is called planar if it can be drawn in the plane in such a way that no two edges cross. Example of a planar graph:.

mickelson
Download Presentation

Planar Graphs

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Great Theoretical Ideas In Computer Science Planar Graphs Lecture 22 CS 15-251

  2. The clique on 4 nodes. A graph is called planar if it can be drawn in the plane in such a way that no two edges cross. Example of a planar graph:

  3. A graph is called planar if it can be drawn in the plane in such a way that no two edges cross. Example of a planar graph: The clique on 4 nodes.

  4. Is K5 planar?

  5. What about K3,3 ?

  6. What about the intriguing object that I brought to class today?

  7. The problem of drawing a graph in the plane arises frequently in VLSI layout problems.

  8. two faces one face Definition: When a graph is drawn in the plane with no crossed edges, this particular embedding of the graph in the plane, it is called a plane graph. A plane graph cuts the plane into regions that we call faces.

  9. Question: Can you redraw this graph as a plane graph so as to alter the number of its faces?

  10. This graph has 6 vertices 8 edges and 4 faces vertices – edges + faces = 2

  11. This graph has 7 vertices 12 edges and 7 faces vertices – edges + faces = 2

  12. Euler 1752 If G is a connected plane graph, then vertices – edges + faces = 2 Let v = # of vertices e = # of edges f = # of faces

  13. Proof: By induction on the # of cycles of G. Base case: G has no cycles. G is connected so it must be a tree. Thus, e = v - 1 and f = 1.

  14. Let v= # of vertices, e= # of edges, f = # of faces exterior e interior By induction hypothesis: Suppose G has at least one cycle C containing edge e. • G is connected since e was on a cycle. • f = f-1 and G has fewer cycles than G. • v= v e= e-1

  15. Corollary: No matter how we redraw a plane graph it will have the same # of faces. Proof: f = 2 – v + e is determined by v and e, neither of which change when we redraw the graph.

  16. A Platonic solid has congruent regular polygons as faces and has the same number of edges meeting at each corner. Platonic Solids Each one can be flattened into a planar graph: With constant degree: k and the same number of edges bounding each face: l

  17. # of edges coming from x vertexx = Each edge belongs to 2 faces: By Euler’s formula: and k,l 3 for physical reasons

  18. The only solutions: tetrahedron cube octahedron dodecahedron icosahedron

  19. Theorem: Every (simple) n-node planar graph G has at most 3n-6 edges. Thus Thus Proof: n = 3: Clearly true. n  3:consider a graph G with a maximal number of edges. • G must be connected or else we could add an edge. Every face has at least 3 edges on its boundary. Every edge lies on the boundary of at most 2 faces.

  20. The Kuratowski Graphs

  21. Corollary: K5 is not planar. A planar graph on n = 5 nodes can have at most 3n-6 = 9 edges. Thus: K5 is not planar.

  22. When we redraw K3,3 , the yellow cycle will be laid out: x a b z y c Fact: K3,3 is not planar either. a x y b z c

  23. Insight 1. If we replace edges in a Kuratowski graph by paths of whatever length, they remain non-planar.

  24. Insight 2 If a graph G contains a subgraph obtained by starting with K5 or K3,3 and replacing edges with paths, then G is non-planar.

  25. Kuratowski’s Theorem [1930] A graph is planar if and only if it contains no subgraph obtainable from K5 or K3,3 by replacing edges with paths.

  26. Appel-Haken Four-Color Theorem [1976] The vertices of any planar graph can be 4-colored in such a way that no two adjacent vertices receive the same color.

  27. Five Color Theorem • Any planar graph can be colored with five colors.

  28. Lemma: Every Planar Graph Contains a Node of Degree · 5 • If every node has degree at least 6, then the number of edges would be 3n, which would contradict our upper bound of 3n-6 edges in an n-node planar graph.

  29. Proof of 5-color theorem • Let G be a node-minimal counter-example to the theorem, i.e., a planar graph that requires 6 colors. • By Lemma, G must have a node q with degree · 5. Let the nodes adjacent to q be named v1, v2, v3, v4, and v5.

  30. V1, V2, V3, V4, V5 can’t form a K5 • Some two neighbors va, and vb of q must not have an edge between them. vb va

  31. Edge Contraction • Contract the edges <q, va> and <q,vb> of G to obtain a planar graph G’. G’ is 5 colorable since it has fewer nodes than G. vb va va, vb, and q become a single node a in G’

  32. Using G’ to 5-color G. • Color va and vb the same as a. Color each node besides q, as it is colored in G’. Color q whatever color is not used on its 5 neighbors. vb va va, vb, and q become a single node a in G’

  33. 2 4 3 User Interface How many objects appear in the pink window? (real-time response required)

  34. (mesh is planar) interior faces Idea: View each rectangle as a mesh.

  35. window border touches no nodes Objects are placed on an nn grid. Data structure will contain: • for each grid node • # objects containing node • for each grid edge • # objects containing edge • for each (interior) face • #objects containing face

  36. ignore object’s nodes and edges outside window Partial Overlap of Window and Object

  37. edge counts in node counts in face counts in # of objects intersecting

  38. sum of node counts in sum of edge counts in sum of face counts in Inclusion-Exclusion Speed-up! store 3-tuple:

  39. a c d b Node count in is a – b – c + d. c a 1 1 0 b d -1 0 0 1 0 1

More Related