Great Theoretical Ideas In Computer Science Planar Graphs Lecture 23 CS 15-251

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.

Is K5 planar?

What about K3,3 ?

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

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

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

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

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

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

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

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.

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

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.

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

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

The only solutions: tetrahedron cube octahedron dodecahedron icosahedron

Theorem: Every (simple) n-node planar graph G has at most 3n-6 edges. 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. Thus Every face has at least 3 edges on its boundary. Thus Every edge lies on the boundary of at most 2 faces.

The Kuratowski Graphs

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.

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

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

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.

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.

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.

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

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

Objects are placed on an n  n grid. window border touches no nodes 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

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

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

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

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