Planar / Non-Planar Graphs

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Planar / Non-Planar Graphs

• Gabriel Laden
• CS146 Spring 2004
• Dr. Sin-Min Lee

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Definitions

• Planar graph that can be drawn without edges
  that intersect within a plane
• Non-Planar graph that cannot be drawn without
  edges that intersect within a plane

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Do Edges Intersect?

• Planar graphs can sometimes be drawn as
  non-planar graphs. It is still a planar graph,
  because they are isomorphic.

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Three Houses / Three Utilities

• Q. Suppose we have three houses and three
  utilities. Is it possible to connect each utility
  to each of three houses without any lines
  crossing?
• Planar or Non-Planar ?
• This is also known as K(3,3) bipartite graph

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Another definition

• Region The area bounded by a subset of the
  vertices and edges of a graph
• Note the outside area of a graph also counts as
  a region. Therefore a tree has one region, a
  simple cycle has two regions.

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Examples of Counting Regions
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Commonly Used Variables

• Variables used in following mathematical proofs
• G an arbitrary graph
• P number of vertices
• Q number of edges
• R number of regions
• n number of edges that bound a region
• N sum of n for all regions of G

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First Theorem

• Let G be a connected planar graph
• p vertices, q edges, r regions
• Then p q r 2
• Theorem is by Euler
• Proof can be made by induction

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Second Theorem

• Let G be a connected planar graph
• p (vertices gt 3), q edges
• Then q lt 3p - 6
• Proof is a little more interesting, uses first
  theorem to help solve

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Proof q lt 3p 6

• For each region in graph, n number of edges to
  form boundary of its region. Sum of all these ns
  in graph N
• N gt 3r must be true, since all regions need at
  least 3 edges to form them.
• N lt 2q must be true, since no edge can be used
  more than twice in forming a region

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(cont) Proof q lt 3p 6

• 3r lt N lt 2q
• Solve p q r 2 for r, then substitute
• 3(-p q 2) lt 2q
• q lt 3p 6 is simplified answer

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Proof K(3,3) is Non-Planar

• Proof by contradiction of theorems
• Since graph is bipartite, no edge connects two
  edges within same subset of vertices
• N gt 4r must be true, since graph contains no
  simple triangle regions of 3 edges.
• N lt 2q must be true, since no edge can be used
  more than twice in forming a region

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(cont) Proof of K(3,3)

• For K(3,3) p6, q 9, r ??
• 4r lt N lt 2q
• 4r lt (2q 2 9 18)
• r lt 4.5
• Using first theorem of planar graphs, p q r
  2
• 6 9 r 2
• r 5
• Proof by contradiction
• r cannot be both equal to 5 and less than 4.5
• Therefore, K(3,3) is a non-planar graph

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Complete Graphs

• Denoted by Kp
• All vertices are connected to all vertices
• q p (p - 1) / 2

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Proof K5 is non-planar

• p5
• q p (p 1) / 2 10
• Using second theorem of planar graphs
• q lt 3p 6
• 10 lt 3(5) 6
• 10 lt 9 ???
• By contradiction, K5 must be non-planar

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More Definitions

• Isomorphic one-to-one maping of two graphs,
  such that they are equivalent
• Subgraph a graph which is contained as part of
  another equivalent or greater graph
• Supergraph if G is a subgraph of G, then G is
  said to be a supergraph of G

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Subdivisions of graph G

• Subdivision a graph obtained from a graph G, by
  inserting vertices of degree two into any edge
• (H is a valid subdivision of G, while F is not)

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Kuratowski Reduction Theorem

• A graph G is planar if and only if G contains no
  subgraph isomorphic to K5 or any sudivision of K5
  or K(3,3)
• Every non-planar graph is a supergraph of K(3,3)
  or K5

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Peterson Graph
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Using Kuratowski

• Q. Is Peterson graph non-planar?
• A. We can use Kuratowski theorem to pick apart
  the graph until we find K5 or K(3,3).
• (solution given on chalkboard)

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Scheduling Problem

• Q. How many time periods are needed to offer the
  following courses for the set of student
  schedules?
• Course Listings
• Combinatorics (C), Graph Theory (G), Linear
  Algebra (L),
• Numerical Analysis (N), Probability (P),
  Statistics (S), Topology(T)
• Student Schedules
• CLT, CGS, GN, CL, LN, CG, NP, GL, CT, CST, PS,
  PT
• A. This can be drawn as a graph, then find the
  chromatic number
• (solution given on chalkboard)

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Chromatic Number Rules

• Four Color Theorem
• If G is a planar graph, then X(G) lt 4
• Theorem for any graph
• Where D(G) max degree of its vertices, X(G) lt
  1 D(G)

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My References