Connection MATH Connection. Benjamin V.C. Collins, James A. Swenson MATH 2730
|
|
- Barbra Goodman
- 5 years ago
- Views:
Transcription
1 MATH 2730 Benjamin V.C. Collins James A. Swenson
2 Traveling Salesman Problem Image: Padberg-Rinaldi, 1987: 532 cities
3 Walks in a graph Let G = (V, E) be a graph. A walk in G is a non-empty list of vertices in which each vertex is adjacent to the next: W = (v 0, v 1,..., v l ) such that v 0 v 1 v l. The number l N is the length of W (even though W contains l + 1 vertices).
4 Walks from a to b The walk W = (v 0, v 1,..., v l ) is called an (a, b)-walk provided a = v 0 and b = v l. Example In the graph below, W = (r, w, u, t, u, s) is an (r, s)-walk.
5 New walks from old The reversal of the walk W = (v 0, v 1,..., v l ) is W 1 = (v l,..., v 1, v 0 ). Let W 1 = (v 0, v 1,..., v l ) and W 2 = (w 0, w 1,..., w k ) be walks. If v l = w 0, then we can define the concatenation W 1 + W 2 = (v 0, v 1,..., v l, w 1,..., w k ), which is a walk of length l + k.
6 Paths in a graph A path is a walk in which no vertex is repeated. An (a, b)-path is a path that begins at a and ends at b. Example In the graph below, W 1 = (r, w, u, t, u, s) is a walk, but not a path; W 2 = (w, r, u, s, t, v) and W 3 = (s) are paths.
7 Why vertices, and not edges? Proposition Let P be a walk in the graph G. If P is a path, then P does not use any edge of G more than once. Proof. We prove the contrapositive. Let P be a walk in G. Suppose that P uses the edge {u, v} at least twice. Then wlog, P = u v u v... or P = u v v u.... In each case, u is repeated in P. Thus P is not a path.
8 Path graphs If V = {v 1,..., v n } and E = {{v k, v k+1 } : 1 k < n}, then (V, E) is called a path graph and denoted by P n. The longest walk in P n has length n 1. P 6
9 Some pairs of vertices are connected Let G = (V, E) be a graph, and let u, v V. We say u is connected to v provided there is a (u, v)-path in G. Example s is connected to t. s is not connected to w. Is s connected to s?
10 as a relation Theorem If G = (V, E) is a graph, then is connected to is an equivalence relation on V. Proof. Define R = {(u, v) V V : u is connected to v}. First, let a V. There is an (a, a)-path in G; namely, (a). Thus ara, so R is reflexive. Now suppose a, b V and arb. Then G contains an (a, b)-path P. Now G contains a (b, a)-path; namely, P 1. Thus bra, so R is symmetric. Finally, let a, b, c V such that arb and brc. Then G contains an (a, b)-path P 1 and a (b, c)-path P 2. The concatenation P 1 + P 2 is an (a, c)-walk in G. Then, for some reason by a lemma, G contains an (a, c)-path. Hence arc, so R is transitive.
11 Walks yield paths Lemma Let G = (V, E) be a graph, and let x, y V. If there is an (x, y)-walk in G, then there is an (x, y)-path in G. Proof. Suppose G contains an (x, y)-walk. By the Well-Ordering Principle, there exists an (x, y)-walk P of minimal length in G. Sftsoc that P is not a path. Let u be a vertex that is repeated in P. Delete a portion of P between two copies of u, including one of the two copies. The result is an (x, y)-walk P in G that is shorter than P, but this is impossible. Thus G contains an (x, y)-path; namely, P. P = (x,..., v j, u, v k,..., y)
12 Components With this lemma, we have finished the proof that is-connected-to is an equivalence relation on V. This means that the vertices of G form equivalence classes: if a V, then [a] = {v V : a is connected to v}. Let G = (V, E) be a graph and let a V. Then [a] V is an equivalence class for the is-connected-to relation, and the induced subgraph G[[a]] is called a component of G.
13 Connected graphs Let G = (V, E) be a graph. We say G is connected provided that any two vertices of G are connected. Proposition A graph G is connected if and only if G has exactly one component.
14 Disconnection Let G = (V, E) be a graph. We say v V is a cut vertex provided that G v has more components than G. We say e E is a cut edge provided that G e has more components than G. Example The graph shown has one cut vertex and one cut edge.
15 How many pieces can you make? Theorem If G is a connected graph and e = {x, y} is a cut edge in G, then H = G e has two components; namely, H[[x]] and H[[y]]. Proof. (See proof in textbook.) Remark If G is a connected graph and v is a cut vertex in G, then the number of components in G may be any number between 2 and d(v).
16 paths Let G be a graph. A path P in G is called provided that P contains every vertex of G. Sir William Rowan Hamilton ( )
17 Knight s Tour of chess
18 A knight s tour found by Euler (1758) E. Sandifer, Euler Did It 30 Knights tour.pdf
19 5 5 knight s tour Proposition A knight s tour exists on a 5 5 chessboard. Proof. a 5 b 5 c 5 d 5 e 5 a 4 b 4 c 4 d 4 e 4 There is a path in the graph shown; namely, a 3 b 3 c 3 d 3 e 3 a 2 b 2 c 2 d 2 e 2 P = (c 3, e 4, c 5, a 4, b 2, d 1, e 3, d 5, b 4, a 2, c 1, e 2, d 4, b 5, a 3, b 1, d 2, c 4, e 5, d 3, e 1, c 2, a 1, b 3, a 5 ). a 1 b 1 c 1 d 1 e 1
20 4 4 knight s tour? Theorem There is no knight s tour on a 4 4 chessboard. Proof. Sftsoc that P is a knight s tour on a 4 4 chessboard. Color the board in various ways, as shown. P must alternate between black and white squares. Sftsoc that P also alternates blue and yellow squares. Then wlog every square visited is blue/white or yellow/black, so the bottom-right square is not visited. So P does not alternate blue and yellow squares. However, it is impossible to move from one yellow square to another. Since there are 4 squares with each pair of colors, the squares of P must be yellow and blue as follows: (y, b, y, b, y, b, y, b, b, y, b, y, b, y, b, y). Likewise, the squares must be red and green in the pattern (r, g, r, g, r, g, r, g, g, r, g, r, g, r, g, r). Now the 8 th and 9 th squares in P are blue/green.
21 That was a lot of work. Remark It is usually fairly easy to show that G has a path (if it does), but extremely difficult to show that G does not have a path (even if it doesn t).
1 Hamiltonian properties
1 Hamiltonian properties 1.1 Hamiltonian Cycles Last time we saw this generalization of Dirac s result, which we shall prove now. Proposition 1 (Ore 60). For a graph G with nonadjacent vertices u and v
More informationMassachusetts Institute of Technology 6.042J/18.062J, Fall 02: Mathematics for Computer Science Professor Albert Meyer and Dr.
Massachusetts Institute of Technology 6.042J/18.062J, Fall 02: Mathematics for Computer Science Professor Albert Meyer and Dr. Radhika Nagpal Quiz 1 Appendix Appendix Contents 1 Induction 2 2 Relations
More informationHamiltonian Graphs Graphs
COMP2121 Discrete Mathematics Hamiltonian Graphs Graphs Hubert Chan (Chapter 9.5) [O1 Abstract Concepts] [O2 Proof Techniques] [O3 Basic Analysis Techniques] 1 Hamiltonian Paths and Circuits [O1] A Hamiltonian
More information8.3 Hamiltonian Paths and Circuits
8.3 Hamiltonian Paths and Circuits 8.3 Hamiltonian Paths and Circuits A Hamiltonian path is a path that contains each vertex exactly once A Hamiltonian circuit is a Hamiltonian path that is also a circuit
More informationSets II MATH Sets II. Benjamin V.C. Collins, James A. Swenson MATH 2730
MATH 2730 Sets II Benjamin V.C. Collins James A. Swenson New sets from old Suppose A and B are the sets of multiples of 2 and multiples of 5: A = {n Z : 2 n} = {..., 8, 6, 4, 2, 0, 2, 4, 6, 8,... } B =
More informationLECTURE 1: INTRODUCTION
LECTURE 1: INTRODUCTION What is a Network? What is a Network? What is the meaning of Network Flows? In our course, a network G=(N,A) consists of a finite number of nodes (in N) which are connected by arcs
More informationAlgorithms and Theory of Computation. Lecture 22: NP-Completeness (2)
Algorithms and Theory of Computation Lecture 22: NP-Completeness (2) Xiaohui Bei MAS 714 November 8, 2018 Nanyang Technological University MAS 714 November 8, 2018 1 / 20 Set Cover Set Cover Input: a set
More informationNgày 20 tháng 7 năm Discrete Optimization Graphs
Discrete Optimization Graphs Ngày 20 tháng 7 năm 2011 Lecture 6: Graphs In this class we shall prove some simple, useful theorems about graphs. We start with very simple obseravations. Lecture 6: Graphs
More informationRelations MATH Relations. Benjamin V.C. Collins, James A. Swenson MATH 2730
MATH 2730 Benjamin V.C. Collins James A. Swenson among integers equals a = b is true for some pairs (a, b) Z Z, but not for all pairs. is less than a < b is true for some pairs (a, b) Z Z, but not for
More informationUnit 7 Day 4 Section 4.5 & Practice Hamiltonian Circuits and Paths
Unit 7 Day 4 Section 4.5 & Practice 4.1-4.5 Hamiltonian Circuits and Paths Warm Up ~ Day 4 D B Is the following graph E 1) Connected? 2) Complete? F G 3) An Euler Circuit? If so, write the circuit. If
More information14 Equivalence Relations
14 Equivalence Relations Tom Lewis Fall Term 2010 Tom Lewis () 14 Equivalence Relations Fall Term 2010 1 / 10 Outline 1 The definition 2 Congruence modulo n 3 Has-the-same-size-as 4 Equivalence classes
More informationContradiction MATH Contradiction. Benjamin V.C. Collins, James A. Swenson MATH 2730
MATH 2730 Contradiction Benjamin V.C. Collins James A. Swenson Contrapositive The contrapositive of the statement If A, then B is the statement If not B, then not A. A statement and its contrapositive
More informationExam EDAF August Thore Husfeldt
Exam EDAF05 25 August 2011 Thore Husfeldt Instructions What to bring. You can bring any written aid you want. This includes the course book and a dictionary. In fact, these two things are the only aids
More informationNP-completeness. Chapter 34. Sergey Bereg
NP-completeness Chapter 34 Sergey Bereg Oct 2017 Examples Some problems admit polynomial time algorithms, i.e. O(n k ) running time where n is the input size. We will study a class of NP-complete problems
More information1.4 Equivalence Relations and Partitions
24 CHAPTER 1. REVIEW 1.4 Equivalence Relations and Partitions 1.4.1 Equivalence Relations Definition 1.4.1 (Relation) A binary relation or a relation on a set S is a set R of ordered pairs. This is a very
More informationBounds on the Traveling Salesman Problem
Bounds on the Traveling Salesman Problem Sean Zachary Roberson Texas A&M University MATH 613, Graph Theory A common routing problem is as follows: given a collection of stops (for example, towns, stations,
More informationCS 320, Fall Dr. Geri Georg, Instructor 320 NP 1
NP CS 320, Fall 2017 Dr. Geri Georg, Instructor georg@colostate.edu 320 NP 1 NP Complete A class of problems where: No polynomial time algorithm has been discovered No proof that one doesn t exist 320
More informationTrees. A tree is a graph which is. (a) Connected and. (b) has no cycles (acyclic).
Trees A tree is a graph which is (a) Connected and (b) has no cycles (acyclic). 1 Lemma 1 Let the components of G be C 1, C 2,..., C r, Suppose e = (u, v) / E, u C i, v C j. (a) i = j ω(g + e) = ω(g).
More informationThe P versus NP Problem. Ker-I Ko. Stony Brook, New York
The P versus NP Problem Ker-I Ko Stony Brook, New York ? P = NP One of the seven Millenium Problems The youngest one A folklore question? Has hundreds of equivalent forms Informal Definitions P : Computational
More informationExam EDAF May 2011, , Vic1. Thore Husfeldt
Exam EDAF05 25 May 2011, 8.00 13.00, Vic1 Thore Husfeldt Instructions What to bring. You can bring any written aid you want. This includes the course book and a dictionary. In fact, these two things are
More informationHamiltonian Cycle. Zero Knowledge Proof
Hamiltonian Cycle Zero Knowledge Proof Hamiltonian cycle Hamiltonian cycle - A path that visits each vertex exactly once, and ends at the same point it started Example Hamiltonian cycle - A path that visits
More informationRing Sums, Bridges and Fundamental Sets
1 Ring Sums Definition 1 Given two graphs G 1 = (V 1, E 1 ) and G 2 = (V 2, E 2 ) we define the ring sum G 1 G 2 = (V 1 V 2, (E 1 E 2 ) (E 1 E 2 )) with isolated points dropped. So an edge is in G 1 G
More informationMATH 22 HAMILTONIAN GRAPHS. Lecture V: 11/18/2003
MATH 22 Lecture V: 11/18/2003 HAMILTONIAN GRAPHS All communities [graphs] divide themselves into the few and the many [i.e., are bipartite]. Alexander Hamilton, Debates of the Federal Convention Before
More informationGraph fundamentals. Matrices associated with a graph
Graph fundamentals Matrices associated with a graph Drawing a picture of a graph is one way to represent it. Another type of representation is via a matrix. Let G be a graph with V (G) ={v 1,v,...,v n
More informationBinomial Coefficients MATH Benjamin V.C. Collins, James A. Swenson MATH 2730
MATH 2730 Benjamin V.C. Collins James A. Swenson Binomial coefficients count subsets Definition Suppose A = n. The number of k-element subsets in A is a binomial coefficient, denoted by ( n k or n C k
More informationThe main limitation of the concept of a. function
Relations The main limitation of the concept of a A function, by definition, assigns one output to each input. This means that a function cannot model relationships between sets where some objects on each
More informationAutomata and Languages
Automata and Languages Prof. Mohamed Hamada Software Engineering Lab. The University of Aizu Japan Mathematical Background Mathematical Background Sets Relations Functions Graphs Proof techniques Sets
More informationJuly 18, Approximation Algorithms (Travelling Salesman Problem)
Approximation Algorithms (Travelling Salesman Problem) July 18, 2014 The travelling-salesman problem Problem: given complete, undirected graph G = (V, E) with non-negative integer cost c(u, v) for each
More informationD1 Discrete Mathematics The Travelling Salesperson problem. The Nearest Neighbour Algorithm The Lower Bound Algorithm The Tour Improvement Algorithm
1 iscrete Mathematics The Travelling Salesperson problem The Nearest Neighbour lgorithm The Lower ound lgorithm The Tour Improvement lgorithm The Travelling Salesperson: Typically a travelling salesperson
More informationRelations. Definition 1 Let A and B be sets. A binary relation R from A to B is any subset of A B.
Chapter 5 Relations Definition 1 Let A and B be sets. A binary relation R from A to B is any subset of A B. If A = B then a relation from A to B is called is called a relation on A. Examples A relation
More informationCS 583: Algorithms. NP Completeness Ch 34. Intractability
CS 583: Algorithms NP Completeness Ch 34 Intractability Some problems are intractable: as they grow large, we are unable to solve them in reasonable time What constitutes reasonable time? Standard working
More informationWarm-up Find the shortest trip (total distance) starting and ending in Chicago and visiting each other city once.
Warm-up Find the shortest trip (total distance) starting and ending in Chicago and visiting each other city once. Minimum-cost Hamiltonian Circuits Practice Homework time Minneapolis Cleveland 779 354
More informationCS/COE
CS/COE 1501 www.cs.pitt.edu/~nlf4/cs1501/ P vs NP But first, something completely different... Some computational problems are unsolvable No algorithm can be written that will always produce the correct
More informationDiscrete Mathematics. Benny George K. September 22, 2011
Discrete Mathematics Benny George K Department of Computer Science and Engineering Indian Institute of Technology Guwahati ben@iitg.ernet.in September 22, 2011 Set Theory Elementary Concepts Let A and
More informationGeneralized knight s tours on rectangular chessboards
Discrete Applied Mathematics 150 (2005) 80 98 www.elsevier.com/locate/dam Generalized knight s tours on rectangular chessboards G.L. Chia, Siew-Hui Ong Institute of Mathematical Sciences, University of
More informationRECAP: Extremal problems Examples
RECAP: Extremal problems Examples Proposition 1. If G is an n-vertex graph with at most n edges then G is disconnected. A Question you always have to ask: Can we improve on this proposition? Answer. NO!
More informationGraphic sequences, adjacency matrix
Chapter 2 Graphic sequences, adjacency matrix Definition 2.1. A sequence of integers (d 1,..., d n ) is called graphic if it is the degree sequence of a graph. Example 2.2. 1. (1, 2, 2, 3) is graphic:
More informationIntro to Contemporary Math
Intro to Contemporary Math Hamiltonian Circuits and Nearest Neighbor Algorithm Nicholas Nguyen nicholas.nguyen@uky.edu Department of Mathematics UK Agenda Hamiltonian Circuits and the Traveling Salesman
More informationMidterm 1. Your Exam Room: Name of Person Sitting on Your Left: Name of Person Sitting on Your Right: Name of Person Sitting in Front of You:
CS70 Discrete Mathematics and Probability Theory, Fall 2018 Midterm 1 8:00-10:00pm, 24 September Your First Name: SIGN Your Name: Your Last Name: Your Exam Room: Name of Person Sitting on Your Left: Name
More information8. INTRACTABILITY I. Lecture slides by Kevin Wayne Copyright 2005 Pearson-Addison Wesley. Last updated on 2/6/18 2:16 AM
8. INTRACTABILITY I poly-time reductions packing and covering problems constraint satisfaction problems sequencing problems partitioning problems graph coloring numerical problems Lecture slides by Kevin
More informationHamiltonian Cycle. Hamiltonian Cycle
Hamiltonian Cycle Hamiltonian Cycle Hamiltonian Cycle Problem Hamiltonian Cycle Given a directed graph G, is there a cycle that visits every vertex exactly once? Such a cycle is called a Hamiltonian cycle.
More informationINF210 Datamaskinteori (Models of Computation)
INF210 Datamaskinteori (Models of Computation) Textbook: John Martin, Introduction to Languages and the Theory of Computation, McGraw Hill, Third Edition. Fedor V. Fomin Datamaskinteori 1 Datamaskinteori
More informationSTRATEGIES OF PROBLEM SOLVING
STRATEGIES OF PROBLEM SOLVING Second Edition Maria Nogin Department of Mathematics College of Science and Mathematics California State University, Fresno 2014 2 Chapter 1 Introduction Solving mathematical
More informationCheck off these skills when you feel that you have mastered them. Write in your own words the definition of a Hamiltonian circuit.
Chapter Objectives Check off these skills when you feel that you have mastered them. Write in your own words the definition of a Hamiltonian circuit. Explain the difference between an Euler circuit and
More informationDefinition of Algebraic Graph and New Perspective on Chromatic Number
Definition of Algebraic Graph and New Perspective on Chromatic Number Gosu Choi February 2019 Abstract This paper is an article on the unusual structure that can be thought on set of simple graph. I discovered
More informationPreliminaries. Graphs. E : set of edges (arcs) (Undirected) Graph : (i, j) = (j, i) (edges) V = {1, 2, 3, 4, 5}, E = {(1, 3), (3, 2), (2, 4)}
Preliminaries Graphs G = (V, E), V : set of vertices E : set of edges (arcs) (Undirected) Graph : (i, j) = (j, i) (edges) 1 2 3 5 4 V = {1, 2, 3, 4, 5}, E = {(1, 3), (3, 2), (2, 4)} 1 Directed Graph (Digraph)
More informationTractable & Intractable Problems
Tractable & Intractable Problems We will be looking at : What is a P and NP problem NP-Completeness The question of whether P=NP The Traveling Salesman problem again Programming and Data Structures 1 Polynomial
More informationWorksheet on Relations
Worksheet on Relations Recall the properties that relations can have: Definition. Let R be a relation on the set A. R is reflexive if for all a A we have ara. R is irreflexive or antireflexive if for all
More informationSAT, Coloring, Hamiltonian Cycle, TSP
1 SAT, Coloring, Hamiltonian Cycle, TSP Slides by Carl Kingsford Apr. 28, 2014 Sects. 8.2, 8.7, 8.5 2 Boolean Formulas Boolean Formulas: Variables: x 1, x 2, x 3 (can be either true or false) Terms: t
More informationMath 2534 Solution to Test 3A Spring 2010
Math 2534 Solution to Test 3A Spring 2010 Problem 1: (10pts) Prove that R is a transitive relation on Z when given that mrpiff m pmod d (ie. d ( m p) ) Solution: The relation R is transitive, if arb and
More informationTopics in Graph Theory
Topics in Graph Theory September 4, 2018 1 Preliminaries A graph is a system G = (V, E) consisting of a set V of vertices and a set E (disjoint from V ) of edges, together with an incidence function End
More informationCircuits. CSE 373 Data Structures
Circuits CSE 373 Data Structures Readings Reading Alas not in your book. So it won t be on the final! Circuits 2 Euler Euler (1707-1783): might be the most prolific mathematician of all times (analysis,
More informationAdvanced Topics in Discrete Math: Graph Theory Fall 2010
21-801 Advanced Topics in Discrete Math: Graph Theory Fall 2010 Prof. Andrzej Dudek notes by Brendan Sullivan October 18, 2010 Contents 0 Introduction 1 1 Matchings 1 1.1 Matchings in Bipartite Graphs...................................
More informationNP-Completeness. CptS 223 Advanced Data Structures. Larry Holder School of Electrical Engineering and Computer Science Washington State University
NP-Completeness CptS 223 Advanced Data Structures Larry Holder School of Electrical Engineering and Computer Science Washington State University 1 Hard Graph Problems Hard means no known solutions with
More informationGreatest Common Divisor MATH Greatest Common Divisor. Benjamin V.C. Collins, James A. Swenson MATH 2730
MATH 2730 Greatest Common Divisor Benjamin V.C. Collins James A. Swenson The world s least necessary definition Definition Let a, b Z, not both zero. The largest integer d such that d a and d b is called
More informationPolynomial-Time Reductions
Reductions 1 Polynomial-Time Reductions Classify Problems According to Computational Requirements Q. Which problems will we be able to solve in practice? A working definition. [von Neumann 1953, Godel
More informationProblem Set 3 Due: Wednesday, October 22nd, 2014
6.89: Algorithmic Lower Bounds Fall 24 Prof. Erik Demaine TAs: Sarah Eisenstat, Jayson Lynch Problem Set 3 Due: Wednesday, October 22nd, 24 Problem. A Tour of Hamiltonicity Variants For each of the following
More informationMATH CIRCLE Session # 2, 9/29/2018
MATH CIRCLE Session # 2, 9/29/2018 SOLUTIONS 1. The n-queens Problem. You do NOT need to know how to play chess to work this problem! This is a classical problem; to look it up today on the internet would
More informationLaceable Knights. Michael Dupuis, Univ. of St. Thomas, St. Paul, Minnesota, USA
Laceable Knights Michael Dupuis, Univ. of St. Thomas, St. Paul, Minnesota, USA dupu3805@stthomas.edu; mdupuis@osii.com; mjd7832@yahoo.com Stan Wagon, Macalester College, St. Paul, Minnesota, USA wagon@macalester.edu,
More informationHW Graph Theory SOLUTIONS (hbovik) - Q
1, Diestel 3.5: Deduce the k = 2 case of Menger s theorem (3.3.1) from Proposition 3.1.1. Let G be 2-connected, and let A and B be 2-sets. We handle some special cases (thus later in the induction if these
More informationInstitute of Operating Systems and Computer Networks Algorithms Group. Network Algorithms. Tutorial 3: Shortest paths and other stuff
Institute of Operating Systems and Computer Networks Algorithms Group Network Algorithms Tutorial 3: Shortest paths and other stuff Christian Rieck Shortest paths: Dijkstra s algorithm 2 Dijkstra s algorithm
More informationHamiltonian circuits in Cayley digraphs. Dan Isaksen. Wayne State University
Hamiltonian circuits in Cayley digraphs Dan Isaksen Wayne State University 1 Digraphs Definition. A digraph is a set V and a subset E of V V. The elements of V are called vertices. We think of vertices
More informationResearch Collection. Grid exploration. Master Thesis. ETH Library. Author(s): Wernli, Dino. Publication Date: 2012
Research Collection Master Thesis Grid exploration Author(s): Wernli, Dino Publication Date: 2012 Permanent Link: https://doi.org/10.3929/ethz-a-007343281 Rights / License: In Copyright - Non-Commercial
More informationRelations. Relations of Sets N-ary Relations Relational Databases Binary Relation Properties Equivalence Relations. Reading (Epp s textbook)
Relations Relations of Sets N-ary Relations Relational Databases Binary Relation Properties Equivalence Relations Reading (Epp s textbook) 8.-8.3. Cartesian Products The symbol (a, b) denotes the ordered
More informationASSIGNMENT 1 SOLUTIONS
MATH 271 ASSIGNMENT 1 SOLUTIONS 1. (a) Let S be the statement For all integers n, if n is even then 3n 11 is odd. Is S true? Give a proof or counterexample. (b) Write out the contrapositive of statement
More informationMalaya J. Mat. 2(3)(2014)
Malaya J Mat (3)(04) 80-87 On k-step Hamiltonian Bipartite and Tripartite Graphs Gee-Choon Lau a,, Sin-Min Lee b, Karl Schaffer c and Siu-Ming Tong d a Faculty of Comp & Mathematical Sciences, Universiti
More informationPartial cubes: structures, characterizations, and constructions
Partial cubes: structures, characterizations, and constructions Sergei Ovchinnikov San Francisco State University, Mathematics Department, 1600 Holloway Ave., San Francisco, CA 94132 Abstract Partial cubes
More informationA Dynamic Programming Approach to Counting Hamiltonian Cycles in Bipartite Graphs
A Dynamic Programming Approach to Counting Hamiltonian Cycles in Bipartite Graphs Patric R. J. Östergård Department of Communications and Networking Aalto University School of Electrical Engineering P.O.
More informationLecture 4: NP and computational intractability
Chapter 4 Lecture 4: NP and computational intractability Listen to: Find the longest path, Daniel Barret What do we do today: polynomial time reduction NP, co-np and NP complete problems some examples
More informationCycles in 4-Connected Planar Graphs
Cycles in 4-Connected Planar Graphs Guantao Chen Department of Mathematics & Statistics Georgia State University Atlanta, GA 30303 matgcc@panther.gsu.edu Genghua Fan Institute of Systems Science Chinese
More informationHAMILTONICITY AND FORBIDDEN SUBGRAPHS IN 4-CONNECTED GRAPHS
HAMILTONICITY AND FORBIDDEN SUBGRAPHS IN 4-CONNECTED GRAPHS FLORIAN PFENDER Abstract. Let T be the line graph of the unique tree F on 8 vertices with degree sequence (3, 3, 3,,,,, ), i.e. T is a chain
More informationReading 11 : Relations and Functions
CS/Math 240: Introduction to Discrete Mathematics Fall 2015 Reading 11 : Relations and Functions Instructor: Beck Hasti and Gautam Prakriya In reading 3, we described a correspondence between predicates
More informationHamiltonian cycles in circulant digraphs with jumps 2, 3, c [We need a real title???]
Hamiltonian cycles in circulant digraphs with jumps,, c [We need a real title???] Abstract [We need an abstract???] 1 Introduction It is not known which circulant digraphs have hamiltonian cycles; this
More informationOptimization Prof. A. Goswami Department of Mathematics Indian Institute of Technology, Kharagpur. Lecture - 20 Travelling Salesman Problem
Optimization Prof. A. Goswami Department of Mathematics Indian Institute of Technology, Kharagpur Lecture - 20 Travelling Salesman Problem Today we are going to discuss the travelling salesman problem.
More informationAlgorithms Design & Analysis. Approximation Algorithm
Algorithms Design & Analysis Approximation Algorithm Recap External memory model Merge sort Distribution sort 2 Today s Topics Hard problem Approximation algorithms Metric traveling salesman problem A
More informationCSC Discrete Math I, Spring Relations
CSC 125 - Discrete Math I, Spring 2017 Relations Binary Relations Definition: A binary relation R from a set A to a set B is a subset of A B Note that a relation is more general than a function Example:
More informationGraphs with few total dominating sets
Graphs with few total dominating sets Marcin Krzywkowski marcin.krzywkowski@gmail.com Stephan Wagner swagner@sun.ac.za Abstract We give a lower bound for the number of total dominating sets of a graph
More informationObservation 4.1 G has a proper separation of order 0 if and only if G is disconnected.
4 Connectivity 2-connectivity Separation: A separation of G of order k is a pair of subgraphs (H 1, H 2 ) so that H 1 H 2 = G E(H 1 ) E(H 2 ) = V (H 1 ) V (H 2 ) = k Such a separation is proper if V (H
More informationOn uniquely 3-colorable plane graphs without prescribed adjacent faces 1
arxiv:509.005v [math.co] 0 Sep 05 On uniquely -colorable plane graphs without prescribed adjacent faces Ze-peng LI School of Electronics Engineering and Computer Science Key Laboratory of High Confidence
More informationA Markov Chain Inspired View of Hamiltonian Cycle Problem
A Markov Chain Inspired View of Hamiltonian Cycle Problem December 12, 2012 I hereby acknowledge valuable collaboration with: W. Murray (Stanford University) V.S. Borkar (Tata Inst. for Fundamental Research)
More informationTraveling Salesman Problem
Traveling Salesman Problem Zdeněk Hanzálek hanzalek@fel.cvut.cz CTU in Prague April 17, 2017 Z. Hanzálek (CTU) Traveling Salesman Problem April 17, 2017 1 / 33 1 Content 2 Solved TSP instances in pictures
More informationGraph Theory. Thomas Bloom. February 6, 2015
Graph Theory Thomas Bloom February 6, 2015 1 Lecture 1 Introduction A graph (for the purposes of these lectures) is a finite set of vertices, some of which are connected by a single edge. Most importantly,
More informationUCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis
UCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis Lecture 10 Class URL: http://vlsicad.ucsd.edu/courses/cse21-s14/ Lecture 10 Notes Midterm Good job overall! = 81; =
More informationTheory of Computer Science. Theory of Computer Science. E5.1 Routing Problems. E5.2 Packing Problems. E5.3 Conclusion.
Theory of Computer Science May 31, 2017 E5. Some NP-Complete Problems, Part II Theory of Computer Science E5. Some NP-Complete Problems, Part II E5.1 Routing Problems Malte Helmert University of Basel
More informationNP and Computational Intractability
NP and Computational Intractability 1 Polynomial-Time Reduction Desiderata'. Suppose we could solve X in polynomial-time. What else could we solve in polynomial time? don't confuse with reduces from Reduction.
More informationGraph Algorithms in Bioinformatics
Graph Algorithms in Bioinformatics Outline 1. Introduction to Graph Theory 2. The Hamiltonian & Eulerian Cycle Problems 3. Basic Biological Applications of Graph Theory 4. DNA Sequencing 5. Shortest Superstring
More informationExam Practice Problems
Math 231 Exam Practice Problems WARNING: This is not a sample test. Problems on the exams may or may not be similar to these problems. These problems are just intended to focus your study of the topics.
More informationCOMP4141 Theory of Computation
COMP4141 Theory of Computation Lecture 4 Regular Languages cont. Ron van der Meyden CSE, UNSW Revision: 2013/03/14 (Credits: David Dill, Thomas Wilke, Kai Engelhardt, Peter Höfner, Rob van Glabbeek) Regular
More information} } } Lecture 23: Computational Complexity. Lecture Overview. Definitions: EXP R. uncomputable/ undecidable P C EXP C R = = Examples
Lecture 23 Computational Complexity 6.006 Fall 2011 Lecture 23: Computational Complexity Lecture Overview P, EXP, R Most problems are uncomputable NP Hardness & completeness Reductions Definitions: P =
More information5. Partitions and Relations Ch.22 of PJE.
5. Partitions and Relations Ch. of PJE. We now generalize the ideas of congruence classes of Z to classes of any set X. The properties of congruence classes that we start with here are that they are disjoint
More informationChapter 9: Relations Relations
Chapter 9: Relations 9.1 - Relations Definition 1 (Relation). Let A and B be sets. A binary relation from A to B is a subset R A B, i.e., R is a set of ordered pairs where the first element from each pair
More informationData Structures and Algorithms
Data Structures and Algorithms CS245-2015S-23 NP-Completeness and Undecidablity David Galles Department of Computer Science University of San Francisco 23-0: Hard Problems Some algorithms take exponential
More informationGraduate Algorithms CS F-21 NP & Approximation Algorithms
Graduate Algorithms CS673-2016F-21 NP & Approximation Algorithms David Galles Department of Computer Science University of San Francisco 21-0: Classes of Problems Consider three problem classes: Polynomial
More informationarxiv:math/ v1 [math.gt] 14 Nov 2003
AUTOMORPHISMS OF TORELLI GROUPS arxiv:math/0311250v1 [math.gt] 14 Nov 2003 JOHN D. MCCARTHY AND WILLIAM R. VAUTAW Abstract. In this paper, we prove that each automorphism of the Torelli group of a surface
More informationChapter 2 - Relations
Chapter 2 - Relations Chapter 2: Relations We could use up two Eternities in learning all that is to be learned about our own world and the thousands of nations that have arisen and flourished and vanished
More informationCS 70 Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand Midterm 1 Solutions
CS 70 Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand Midterm 1 Solutions PRINT Your Name: Answer: Oski Bear SIGN Your Name: PRINT Your Student ID: CIRCLE your exam room: Dwinelle
More informationMATH 2200 Final Review
MATH 00 Final Review Thomas Goller December 7, 01 1 Exam Format The final exam will consist of 8-10 proofs It will take place on Tuesday, December 11, from 10:30 AM - 1:30 PM, in the usual room Topics
More informationInternational Journal of Trend in Research and Development, Volume 3(5), ISSN: Hamiltonian and Eulerian Cycles
Hamiltonian and Eulerian Cycles Vidhi Sutaria M.Tech in Information and Network Security, Computer Science and Engineering Department, Institute of Technology, Nirma University, Ahmadabad, Gujarat, India
More informationDesign and Analysis of Algorithms
Design and Analysis of Algorithms CSE 5311 Lecture 25 NP Completeness Junzhou Huang, Ph.D. Department of Computer Science and Engineering CSE5311 Design and Analysis of Algorithms 1 NP-Completeness Some
More informationGraph G = (V, E). V ={vertices}, E={edges}. V={a,b,c,d,e,f,g,h,k} E={(a,b),(a,g),( a,h),(a,k),(b,c),(b,k),...,(h,k)}
Graph Theory Graph G = (V, E). V ={vertices}, E={edges}. a b c h k d g f e V={a,b,c,d,e,f,g,h,k} E={(a,b),(a,g),( a,h),(a,k),(b,c),(b,k),...,(h,k)} E =16. Digraph D = (V, A). V ={vertices}, E={edges}.
More information