Three Houses-Three Utilities Problem

Slides:

Advertisements

Similar presentations

Certifying algorithms Algorithms and Networks. Certifying algorithms What is it? Examples: –Euler tour –Bipartite graphs –Flow –Planarity Certifying algorithms2.

Advertisements

Planar / Non-Planar Graphs Gabriel Laden CS146 – Spring 2004 Dr. Sin-Min Lee.

Graph Theory Chapter 9 Planar Graphs 大葉大學資訊工程系黃鈴玲.

Kerry R. Sipe Aug. 1, 2007 On the Non-Orientable Genus of Zero Divisor Graphs Missouri State University REU.

Week 11 – Cop number of outerplanar graphs Dr. Anthony Bonato Ryerson University AM8002 Fall 2014.

Tucker, Applied Combinatorics, Section 1.4, prepared by Patti Bodkin

2IL90: Graph Drawing Introduction Fall Graphs  Vertices  Edges.

Drawing of G. Planar Embedding of G Chord A chord of a cycle C is an edge not in C whose endpoints lie in C.

Drawing of G. Planar Embedding of G Proposition Proof. 1. Consider a drawing of K 5 or K 3,3 in the plane. Let C be a spanning cycle. 2. If the.

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 4, Monday, September 8.

Curve Curve: The image of a continous map from [0,1] to R 2. Polygonal curve: A curve composed of finitely many line segments. Polygonal u,v-curve: A polygonal.

Problem: Induced Planar Graphs Tim Hayes Mentor: Dr. Fiorini.

Matthew Bowling Euler’s Theorem Mathfest Spring ‘15.

Graph Theory Ch6 Planar Graphs. Basic Definitions  curve, polygon curve, drawing  crossing, planar, planar embedding, and plane graph  open set  region,

Graph Theory Chapter 6 Planar Graphs Ch. 6. Planar Graphs.

Planar Graphs: Euler's Formula and Coloring Graphs & Algorithms Lecture 7 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.:

Mathematics of Networks (Cont)

MAT 2720 Discrete Mathematics Section 8.7 Planar Graphs

VLSI Layout Algorithms CSE 6404 A 46 B 65 C 11 D 56 E 23 F 8 H 37 G 19 I 12J 14 K 27 X=(AB*CD)+ (A+D)+(A(B+C)) Y = (A(B+C)+AC+ D+A(BC+D)) Dr. Md. Saidur.

Stefan Felsner planar graph G planar embedding of G A graph G is planar if it can be drawn on the plane with no crossing edges.

Graph theory and networks. Basic definitions  A graph consists of points called vertices (or nodes) and lines called edges (or arcs). Each edge joins.

x y Example 1 x y X = 3.

MAT 2720 Discrete Mathematics Section 8.2 Paths and Cycles

Chapter 9.7 Planar Graphs These class notes are based on material from our textbook, Discrete Mathematics and Its Applications, 6th ed., by Kenneth H.

Indian Institute of Technology Kharagpur PALLAB DASGUPTA Graph Theory: Planarity Pallab Dasgupta, Professor, Dept. of Computer Sc. and Engineering, IIT.

Planar Graphs prepared and Instructed by Arie Girshson Semester B, 2014 June 2014Planar Graphs1.

Chapter 7 Planar Graphs 大葉大學資訊工程系黃鈴玲  7.2 Planar Embeddings  7.3 Euler’s Formula and Consequences  7.4 Characterization of Planar Graphs.

Planar Graphs (part 2) prepared and Instructed by Gideon Blocq Semester B, 2014 June 2014Planar Graphs (part 2)1.

Graphs. Representations of graphs ： undirected graph An undirected graph G have five vertices and seven edges An adjacency-list representation of G The.

Planar Graphs Hubert Chan (Chapter 9.7) [O2 Proof Techniques]

Outline 1 Properties of Planar Graphs 5/4/2018.

Discrete Mathematics Graph: Planar Graph Yuan Luo

Characteristics of Planar Graphs

Applied Combinatorics, 4th Ed. Alan Tucker

Vector representations of graphs

If tomorrow is Thursday, then today is Wednesday.

Warm up Solve the inequality. Then graph the solution.

Chapter 3 Section 5.

Chapter 10.7 Planar Graphs These class notes are based on material from our textbook, Discrete Mathematics and Its Applications, 7th ed., by Kenneth H.

Planarity and Euler’s Formula

Planarity Cox, Sherman, Tong.

The Implicit Function Theorem---Part 1

Chapter 10.7 Planar Graphs These class notes are based on material from our textbook, Discrete Mathematics and Its Applications, 7th ed., by Kenneth H.

Chapter 10.7 Planar Graphs These class notes are based on material from our textbook, Discrete Mathematics and Its Applications, 7th ed., by Kenneth H.

Applied Combinatorics, 4th Ed. Alan Tucker

Chapter 10.7 Planar Graphs These class notes are based on material from our textbook, Discrete Mathematics and Its Applications, 7th ed., by Kenneth H.

Chapter 10.7 Planar Graphs These class notes are based on material from our textbook, Discrete Mathematics and Its Applications, 7th ed., by Kenneth H.

MAT 2720 Discrete Mathematics

Chapter 10.7 Planar Graphs These class notes are based on material from our textbook, Discrete Mathematics and Its Applications, 8th ed., by Kenneth H.

N(S) ={vV|uS,{u,v}E(G)}

GRAPHS Lecture 17 CS2110 Spring 2018.

Gaph Theory Planar Graphs

Function or Not? 3.06.

GRAPH THEORY Properties of Planar Graphs Ch9-1.

Chapter 10.7 Planar Graphs These class notes are based on material from our textbook, Discrete Mathematics and Its Applications, 7th ed., by Kenneth H.

Drawing a graph

Drawing a graph

Agenda Review Lecture Content: Shortest Path Algorithm

e2 e1 e5 e4 e3 v1 v2 v3 v4 f The dimension of f =

QUESTION 9 The given inequality is

Chapter 10.7 Planar Graphs These class notes are based on material from our textbook, Discrete Mathematics and Its Applications, 7th ed., by Kenneth H.

GRAPH THEORY Properties of Planar Graphs Ch9-1.

Drawing a graph

Presentation transcript:

Three Houses-Three Utilities Problem Connect each house to each utility... When drawing, the lines cannot cross!

Three Houses-Three Utilities Problem

Three Houses-Three Utilities Problem ?

Three Houses-Three Utilities Problem You Can't Do It!

Formalize The Problem: Graphs and Planarity

Formalize The Problem: Graphs and Planarity

Formalize The Problem: Graphs and Planarity

Formalize The Problem: Graphs and Planarity is a minor of Subgraph Edge to Contract After Contraction

Another Special Graph... Can't be drawn in the plane

Makings of a Theorem Every graph which contains one of theses as a minor is not embeddable.

Makings of a Theorem Conversely, every graph which not embeddable contains one of these graphs! If I know this is not planar... Then I can find one of these

Minors and Kuratowski's Theorem

A Relevant Graph Property: Outerplanarity Not Outerplanar

A Relevant Graph Property: Outerplanarity

(as drawn in most textbooks) A Way of Viewing nonplanarity from nonouterplanarity our nonplanar graphs (as drawn in most textbooks) “violations” of our outerplanar graphs rearrange them

Mixing Things Up: Planarity On Different Manifolds Fundamental Domain Of Torus:

Mixing Things Up: Planarity On Different Manifolds Fundamental Domain Of Torus:

Mixing Things Up: Planarity On Different Manifolds

Mixing Things Up: Planarity On Different Manifolds

Mixing Things Up: A General Statement and A General Answer

A General Phenomena in Graphs

A General Phenomena in Graphs

A General Phenomena in Graphs

A General Theorem for Graphs!

What Next? Abstract Simplicial Complexes

What Next? Abstract Simplicial Complexes

Differences Start To Become Apparent... We need to change some things up: Link Condition Up to an equivalence relation Piecewise Linearity

Alexander’s Horned Sphere Differences Start To Become Apparent... Theorem does not hold in higher dimensions. We will require that our embeddings be Piecewise-Linear so that a higher dimensional Schoenflies holds. Alexander’s Horned Sphere

Goal: Get Planarity Criterion via Minors Try Our First Strategy: Outerplanarity to Planarity Can we characterize outerplanar complexes? no... we have problems do to multiply connected regions

Goal: Get Planarity Criterion via Minors In addition we have nonorientable complexes: the real projective plane

Goal: Get Planarity Criterion via Minors 2.0 Try Our Second Strategy: Outerplanarity to Planarity with modified hypothesis: All Cycles Are Spheres. What can we say? If all the vertices lie in some cycle

Goal: Get Planarity Criterion via Minors 2.0 Nonplanar are “violations” of our outerplanar graphs What we want to say...

Goal: Get Planarity Criterion via Minors 2.1 Try Our Second Strategy: Outerplanarity to Planarity with modified hypothesis This is FALSE, there are complexes which are not planar, but not even nonouterplanar before they become outerplanar coning a nonplanar graph

Goal: Get Planarity Criterion via Minors 3.0 Try Our Second Strategy: Outerplanarity to Planarity with modified hypothesis and demanding that it has an outerplanar subgraph What we want to say... That our characterization of outerplanarity is correct. Can we even restore the hypothesis for cycles to get a characterization of outerplanarity? NO... we have a counterexample

Goal doesn't looks so good... Note: our inequalities for determining non-planarity go the wrong way in odd dimensions.

On The Other Hand... What can we say...

On The Other Hand... Linkless Embeddability Relationship with Linkless embeddability