Mathematics for Computer Science

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Mathematics for Computer Science

• Lecture 10 Graphs (3)
• Leon van der Torre

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Slides

• Slides can be found on the internet
• These slides are based on MIT open courseware,
  by A. Meyer

BECS - Bachelor of Engineering in Computer
Science
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Connections, not Layout
Same graph, different layouts
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Isomorphic Graphs
Same graph, different labels
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Graph Isomorphism

• Graphs G1 and G2 are isomorphic
• there is an exact correspondence
• between vertices of G1 and G2 that
• preserves edges.
• ? bijection f V1 ? V2
• uv is in E1 iff f(u)f(v) is in E2

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Are these Isomorphic?
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Show a Bijection
bijection f(Dog) beef f(Cat) tuna f(Cow)
hay f(Pig) corn
Check edge preservation
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Finding the Mapping

• Not easy --many possible mappings
• Can test for properties
• preserved under isomorphism
• of nodes, edges
• degree distributions
• length of paths cycles

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Non-isomorphism
degree 2
all degree 3
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Exercise

• Define an isomomorphism between the graphs, or
  prove that there is none.
• V1 1,2,3,4,5,6 , E1 12, 23, 34, 14, 15,
  35, 45
• V2 1,2,3,4,5,6 , E2 12, 23, 34, 45, 51,
  24, 25
• V1 1,2,3,4,5,6 , E1 12, 23, 34, 14, 45,
  56, 26
• V2 a,b,c,d,e,f , E2 ab, bc, cd, de, ae,
  ef, cf
• V1 a,,h , E1 ab, bc, cd, ad, ef, fg, gh,
  he, dh, bf
• V2 s,, z , E2 st, tu, uv, sv,wx, xy,
  yz,wz, sw, vz

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Possible Graph?
Is there a graph with vertex degrees 2,2,1?
NO!
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An Impossible Graph
221 odd

• The Handshaking Lemma
• sum of degrees is even.

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The Handshaking Lemma
Proof Each edge contributes 2 to the sum on the
right
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Exercise

• There is a party. Some people shake hands an even
  number of times and some shake an odd number of
  times.
• Proposition an even number of people shake hands
  an odd number of times.
• Represent the party as a graph and the
  proposition as graph theoretical property.
• Prove the proposition.

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Paths Simple Paths

• Path sequence of vertices

Simple path all different
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Connectedness

• vertices v, w are connected iff
• there is a path starting at v and
• ending at w.
• A graph is connected iff every two of its
  vertices are connected

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Paths Simple Paths

• Lemma
• The shortest path connecting
• two vertices is simple!

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Connected Components

• The more connected components,
• the more broken up" the graph is.

3 connected components
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Connected Components

• The connected component
• of vertex v

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Connected Components

• A graph is connected
• iff it has only
• 1 connected component

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Cycles

• Cycle path from v0 to v0

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Trees

• A tree is a connected graph
• with no simple cycles.

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some 5-vertex trees
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Applications of Trees

• Data structures for sorting, searching
• Spanning Trees
• Game Trees (alpha-beta trees)
• Pre?x codes (Huffman encoding)
• Many algorithms based on trees

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Tree Isomorphisms
same
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Equivalent Tree Definitions

• lemma A tree is a graph with a unique path
  between any 2 vertices.
• lemma A tree is a connected graph with
  vertices 1 edges.

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Be careful

• ?A tree is a graph with 1 more vertex than edges?
• NO

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the only 5-vertex Trees
Self-test show every 5-vertex tree isomorphic to
one of these
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Solution to Self-test

• 5-vertex tree has 4 edges
• so sum of degrees is 8
• tree is connected
• so vertex degrees are gt 0
• leaves 3 degrees to distribute
• The only possibilities are
• 3 2 1 1 1
• 2 2 2 1 1
• 4 1 1 1 1
• these force unique connections

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Spanning Trees

• a spanning tree of a connected graph is a tree
• obtained by deleting edges
• Always exists
• subgraph with minimum edges
• connecting all vertices

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Spanning tree example
graph courtesy of Steven Rudich, CMU
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Flight Gates

• flights need gates, but times overlap.
• how many gates needed?

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Airline Schedule
122 145 67 257 306 99
Flights
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Model as a Graph
145
306
99
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Model as a Graph
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Color vertices

• so adjacent vertices have
• different colors.
• colors gates needed

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Coloring the Vertices
assign gates
4 colors 4 gates
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Better coloring
3 colors 3 gates
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Final Exams

• subjects conflict if student
• takes both, so
• need different time slots.
• how short an exam period?

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Model as a Graph
8.02
6.042
18.02
assign times
3.091
4 time slots (best possible)
6.001
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Map Coloring
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Four Color Theorem
any planar map is 4-colorable. 1850s false
proof published (was correct for 5
colors). 1970s prf with much computing 1990s
much improved
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Chromatic Number

• min colors for G is
• chromatic number, ?(G)
• lemma
• ?(tree) 2

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Trees are 2-colorable

• Pick any vertex as root.
• if (unique) path from root is
• even length
• odd length

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Simple Cycles
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Complete Graph K5
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Bounded Degree

• if all vertex degrees k, then

by simple recursive coloring procedure
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- Prove that it can be colored with 3 colors -
Prove that it can not be colored with 2 colors
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Exercise

• A portion of a computer program consists of a
  sequence of calculations where the results are
  stored in variables
• Inputs a, b
• 1. c a b
• 2. d a c
• 3. e c 3
• 4. f c - e
• 5. g a f
• 6. h f 1
• Outputs d, g, h

(a) Recast the register allocation problem as a
question about graph coloring. What do the
vertices correspond to? Under what conditions
should there be an edge between two vertices?
Construct the graph corresponding to the
example. (b) Color your graph using as few colors
as you can. Call the computers registers R1, R2,
etc. Describe the assignment of variables to
registers implied by your coloring. How many
registers do you need?
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Compatible Boys Girls
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Compatible Boys Girls
match each girl to a unique compatible boy
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Compatible Boys Girls
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Compatible Boys Girls
B
G
suppose this edge was missing
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Compatible Boys Girls
no match possible
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No match possible
because of bottleneck
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Bottleneck condition
B
G
N(S)2
S3
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Bottleneck Lemma
bottleneck not enough boys for some set of girls.

• If there is a bottleneck, then no match is
  possible.

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Halls Theorem

• There is a perfect match iff there are no
  bottlenecks.
• Proof clever strong induction on girls.
• (Better proof using duality principle
• goes beyond course)

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Halls Theorem

• There is a perfect match iff there are no
  bottlenecks.
• Lots of elegant uses in applications proofs

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Planar Graphs
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Planar Graphs
A graph is planar if there is a way to draw it in
the plane without edges crossing.
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Planar Graphs
Maps are 2-connected planar graphs

• General connected planar graphs may have
• dongles

cross bars
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A Planar Graph
(wait! also the outer face)
with 3 faces
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4
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Planar Graphs
draw it edge by edge
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Planar Graphs
and record faces while drawing
graph
faces
(the outer face)
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Planar Graphs
and record faces while drawing
graph
faces
(the outer face)
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Planar Graphs
and record faces while drawing
graph
faces
(the outer face)
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If you like curves

• With same faces, you can draw the graph in the
  plane big or small, curvy or straight

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2
1
3
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Planar Drawing Faces

• A planar embedding is a graph and its faces.
• The same planar graph may have different
  embeddings.

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Recursive Def Planar Embeddings

• Base a graph consisting of a single vertex, v,
  and no edges, along with a single face consisting
  of the length 0 path from v to v,
• is a PE.

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Adding an edge to an embedding

• Two constructor cases
• Attach edge from vertex on a face to a new
  vertex.
• Attach edge between vertices on a face.

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Face Creation Rules
1) choose face
add edge to new vertex b
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Face Creation Rules
1) choose face
new face
add edge to new vertex b
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Face Creation Rules
1) choose face
nothing else changes
new face
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Face Creation Rules
2) choose face
add edge across it
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Face Creation Rules
2) choose face
splits into 2 faces
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Face Creation Rules
2) choose face
b
b
a
a
splits into 2 faces
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Euler's Formula
If a planar embedding has v vertices, e edges,
and f faces, then v e f 2
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Euler's Formula

• Proof by structural induction on embeddings
• base case 1 vertex

v 1, f 1, e 0 1?0?1 2
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Adding an edge to a drawing

• Inductive step
• Case 1 edge to new vertex
• v increases by 1
• e increases by 1
• f stays the same
• so v e f stays the same

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Adding an edge to a drawing

• Inductive step
• Case 2 edge across face
• v stays the same
• e increases by 1
• f increases by 1
• so v e f stays the same
• Inductive steps

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Homework

• Outils mathématiques pour l'informaticien,
• Section 6.3