Solutions to Exercises Chapter 11: Graphs

Save this PDF as:

Size: px
Start display at page:

Transcription

1 Solutions to Exercises Chapter 11: Graphs 1 There are 34 non-isomorphic graphs on 5 vertices (compare Exercise 6 of Chapter 2). How many of these are (a) connected, (b) forests, (c) trees, (d) Eulerian, (e) Hamiltonian, (f) bipartite? The numbers are (a) 21; (b) 10; (c) 3; (d) 4; (e) 8; (f)13. (The answer in the book for (a) is wrong; there are 13 disconnected graphs.) It is probably simplest to list all 34 graphs and check the six properties. For jobs of this kind the Atlas of Graphs (ed. R. C. Read and R. J. Wilson), Oxford University Press, 1998, is useful. 2 Show that the Petersen graph (Section 11.12) is not Hamiltonian, but does have a Hamiltonian path. In the drawing of the Petersen graph in Figure 11.4 on page 183, we distinguish five edges of the outer pentagon, five edges of the inner pentagram, and five crossing edges. A Hamiltonian circuit must return to its starting point, and so must use an even number (2 or 4) of crossing edges. If there are two crossing edges, then their ends in both the outer pentagon and the inner pentagram must be joined by paths of length 4. This implies that these ends must be adjacent in both the outer pentagon and the inner pentagram, which is impossible. If there are four crossing edges, then we must have three edges in both the outer pentagon and the inner pentagram, including two through each of the points not on chosen crossing edges. There is a unique such configuration, but it consists of two 5-cycles rather than a Hamiltonian circuit. A Hamiltonian path is easily found: follow a path containing four edges of the outer pentagon, then take a crossing edge, then follow four edges of the inner pentagram. 3 Show that the greedy algorithm does not always succeed in finding the path of least weight between two given vertices in a connected edge-weighted graph. It is not completely clear what is meant by the greedy algorithm in this case. If we have to construct a path from x to y, we take the algorithm as follows: start at x, and at any stage, take that edge of smallest weight whose other end is nearer 1

2 to y than the current vertex. If x and y are joined by two paths of length 2, with edge weights 3 and 3 on one path, 4 and 1 on the other, then this algorithm will fail. (For this example, it would succeed in finding a shortest path from y to x. If, however, we take x and y to be joined by two paths of length 3, with edge weights 2,1,2 on one and 1,4,1 on the other, then it fails in both directions.) 4 Consider the modification of the greedy algorithm for minimal connector. Choose the edge e for which w(e) is minimal subject to the conditions that S + e contains no cycle and e shares a vertex with some previously chosen edge (unless S = /0). Prove that the modified algorithm still correctly finds a minimal connector. We suppose that all the edge weights are different. Then there is a unique minimal spanning tree T, found by GA (the greedy algorithm, as described in the text). If we apply MGA (the modified greedy algorithm, as in the question) to T, ignoring all other edges, it must choose all the edges of T in some order, say e 1,e 2,...,e n 1. Suppose that, when applied to the whole graph G, the MGA doesn t choose T. Suppose i is minimal with respect to the property that it doesn t choose e i, but chooses another edge f instead. (Note that i > 1, since e 1 is the edge of smallest weight.) The edges {e 1,...,e i 1 } form a subtree T, and f is chosen as the edge of smallest weight joining T to a new vertex. There is a unique circuit in T { f }, containing a unique edge (other than f ) joining T to a vertex outside it, say e j. Then w( f ) < w(e j ). But if we delete e j from T and replace it with f, we obtain a tree of smaller weight, contradicting the minimality of T. If we allow some edge weights to vary continuously so that they become equal to other edge weights, both the weight of a minimal connector and the weight produced by MGA vary continuously, and so they remain equal in this case. 5 Let G = (V,E) be a multigraph in which every vertex has even valency. Show that it is possible to direct the edges of G (that is, replace each unordered pair {x,y} by the ordered pair (x,y) or (y,x)) so that the in-valency of any vertex is equal to its out-valency. By (11.4.1), each connected component of G is Eulerian. Take a closed Eulerian trail in each component, and direct the edges according to their direction of transit while following this trail. Now we must leave each vertex as often as we enter it; so the in- and out-valency of any vertex in the directed graph are equal. 2

3 6 Let G be a graph on n vertices. Suppose that, for all non-adjacent pairs x,y of vertices, the sum of the valencies of x and y is at least n 1. Prove that G is connected. Let x and y be two vertices. If they are adjacent, there is a path of length 1 between them; so suppose not. Let X and Y be the sets of neighbours of x and y respectively. Then X + Y n 1. But X Y V (G)\{x,y}; so X Y n 2. Hence X Y 1, and there is a path of length 2 from x to y (via a vertex in this intersection). So G is connected (and has diameter at most 2). Remark. This is best possible for n even, the disjoint union of two complete graphs of size n/2 demonstrates this. Note that strengthening the bound by just one forces the graph to be Hamiltonian (Ore s Theorem). 7 (a) Prove that a connected bipartite graph has a unique bipartition. (b) Prove that a graph G is bipartite if and only if every circuit in G has even length. (a) If G is connected, then two points lie in the same bipartite block if and only if the length of a path joining them is even. This determines the bipartition uniquely. (b) Since a graph is bipartite if and only if each connected component is bipartite, it is enough to prove this for a connected graph. Now if a connected bipartite graph contains an odd circuit, then a vertex on this circuit is joined to itself by a path of odd length, and so lies in the opposite bipartite block to itself, a contradiction. Conversely, suppose that G is connected, and that all its circuits have even length. Then, for any vertices x and y, the parity of all paths joining x and y is the same (else an odd circuit would be formed). So, given a vertex x, we may divide the graph into the sets of vertices joined to x by even, resp. odd, paths. Any edge must join vertices in different sets, so this is a bipartition. 8 Choose ten towns in your country. Find from an atlas (or estimate) the distances between all pairs of towns. Then (a) find a minimal connector; (b) use the twice-round-the-tree algorithm to find a route for the Travelling Salesman. How does your route in (b) compare with the shortest possible route? The answer clearly depends on the towns chosen. 3

4 The most certain way to establish the shortest possible route for the Travelling Salesman is by exhaustive search. Keeping the first town fixed, permute the remaining nine in all possible ways (using the algorithm of (3.12.4)); for each permutation, calculate the length of the circuit, and record the shortest length. A computer will perform this calculation in a few seconds. 9 Consider the result of Chapter 6, Exercise 7, viz. Let F = (A 1,...,A n ) be a family of sets having the property that A(J) J d for all J {1,...,n}, where d is a fixed positive integer. Then there is a subfamily containing all but d of the sets of F, which possesses a SDR. Prove this by modifying the proof of Hall s Theorem from König s given in the text. Following the proof of ( ), we see that an edge-cover S satisfies S n d; by König s Theorem, there is a matching of size (at least) n d, that is, a SDR for all but d of the sets. For the converse, assume that the version of Hall s Theorem given in this question is valid, and let G be a bipartite graph with bipartite blocks of sizes m and n, and with smallest edge-cover of size k. Clearly there cannot be a matching of size larger than k; we have to find one of size k. Let the vertices in the two bipartite blocks be X = {v 1,...,v m } and Y = {w 1,...,w n }, and let A i be the set of neighbours of w i. Given J {1,...,n}, let W(J) = {w j : j J}. Then A(J) (Y \W(J)) is an edge-cover; and so A(J) + n J k, whence A(J) J d, where d = n k. By Hall s Theorem, there is a set J, with J = n d, for which the family (A j : j J) has an SDR, say (x j : j J). Then the edges {x j,y j }, for j J, form a matching, of size n d = k. 10 König s Theorem is often stated as follows: The minimum number of lines (rows or columns) which contain all the non-zero entries of a matrix A is equal to the maximum number of independent non-zero entries, where a set of matrix entries is independent if no two are in the same row or column. Show the equivalence of this form with the one given in the text. 4

5 Assume König s Theorem in the bipartite graph form, and let A be a matrix. Form a graph as described in the question. Now a non-zero matrix entry corresponds to an edge of the graph, and a set of independent such entries to a set of disjoint edges, that is, a matching. Also, a set of lines containing all non-zero entries translates to a set of vertices incident with all edges, that is, an edge-cover. So the matrix form of the theorem follows. Conversely, assume the matrix form, and let G be a bipartite graph with bipartite blocks {v 1,...,v m } and {w 1,...,w n }. Let A be the m n matrix with (i, j) entry 1 if {v i,w j } is an edge of G, and 0 otherwise. Now the translation reverses: a matching in G is a set of independent non-zero entries of A, and an edge-cover is a set of lines containing all non-zero entries. So the bipartite graph form follows. 5

6 11 In this exercise, we translate the stepwise improvement algorithm in the proof of the Max-Flow Min-Cut Theorem into an algorithm for König s Theorem. Let G be a bipartite graph with bipartition {A,B}. We observed in the text that an integer-valued flow f in N(G) corresponds to a matching M in G, consisting of those edges {a,b} for which the flow in (a,b) is equal to 1. Now consider the algorithm in the proof of the Max-Flow Min-Cut Theorem, which either increases the value of the flow by 1, or finds a cut. Suppose that we are in the first case, where there is a path (s,a 1,b 1,a 2,b 2,...,a r,b r,t) in the underlying graph of N(G) along which the flow can be increased. Then (a 1,b 1,...,a r,b r ) is a path in G, such that all the edges {b i,a i+1 } but none of the edges {a i,b i } belong to M; moreover, no edge containing a 1 or b r is in M. Such a path in G is called an alternating path with respect to M. (An alternating path starts and ends with an edge not in M, and edges not in and in M alternate. Moreover, since no edge of M contains a 1 or b r, it cannot be extended to a longer such path.) Show that, if we delete the edges {b i,a i+1 } from M (i = 1,...,r 1), and include the edges {a i,b i } (i = 1,...,r), then a new matching M with M = M + 1 is obtained. So the algorithm is: WHILE there is an alternating path, apply the above replacement to find a larger matching. When no alternating path exists, the matching is maximal. The question doesn t actually ask you to do anything! But here is a set-up which both allows efficient search for an alternating path, and also allows a simple proof from König s Theorem that the matching is maximum if no alternating path exists. Let G be a bipartite graph with bipartition {A,B}, and let M = {{a 1,b 1 },...,{a r,b r }} be a matching in G. We form an auxiliary digraph A(M) as follows. The vertices of A(M) are symbols e 1,...,e r corresponding to the edges of M, and two new 6

7 symbols s and t. The edges in A(M) are as follows: (e i,e j ), if there is an edge {a i,b i+1 } in G; (s,e i ), if there is an edge {a,b i } in G, where a is on no edge of M; (e j,t), if there is an edge {a j,b} in G, where b is on no edge of M. Now, if there is a directed path from s to t in A(M), then it arises from an alternating path in G, and so a larger matching exists. Suppose that no directed path from s to t exists in A(M). Let S be the set of vertices reachable from s in A(M), and T the remaining vertices. Then, if e i S and e j T, then there is no edge from a i to b j, or from a to b j with a not covered by M, or from a i to b with b not covered by M. It follows that {b i : e i S} {a j : e j T } is an edge-cover of G. Its cardinality is equal to that of M; hence no larger matching can exist. 12 Let G be a graph with adjacency matrix A. Prove that the (i, j) entry of A d is equal to the number of walks of length d from i to j. The (i, j) entry of A d is equal to k 1,...,k d 1 a ik1 a k1 k 2 a kd 1 j. A term in this sum is zero unless {i,k 1 }, {k 1,k 2 },..., {k d 1, j} are all edges (that is, (i,k 1,k 2,...,k d 1, j) is a walk of length d from i to j), in which case it is 1. So the sum counts the number of walks of length d from i to j. 7

8 13 This exercise proves the friendship theorem : in a finite society in which any two members have a unique common friend, there is somebody who is everyone else s friend. In graph-theoretic terms, a graph on n vertices in which any two vertices have exactly one common neighbour, possesses a vertex of valency n 1, and is a windmill. Step 1 Let the vertices be 1,...,n, and let A i be the set of neighbours of i. Using the de Bruijn Erdős Theorem (Chapter 7), or directly, show that either there is a vertex of valency n 1, or all sets A i have the same size (and the graph is regular). In the latter case, the sets A i are the lines of a projective plane (Chapter 9). Step 2 Suppose that G is regular, with valency k. Use the eigenvalue technique of Section to prove that k = 2. (a) Using the de Bruijn Erdős Theorem: Let A i be the set of neighbours of j. By assumption, A j A k = 1 for all j k. Hence either all the sets A j have the same cardinality k (so that the graph is regular) or, say A 1 = 1 and A j = 2 for j 1. In the latter case, vertex 1 is joined to all others, and the remaining vertices pair off as in the figure. Directly: We show that two non-adjacent vertices x and y have the same valency. There is one vertex z joined to x and y; any other neighbour of x is joined to a unique neighbour of y, and vice versa. Suppose that G is not regular. Any two vertices with different valency are joined. If A is the set of all vertices with valency a, then either A contains just one vertex, or two vertices in A have n A common neighbours at least, so that A contains all but one vertex. For n > 2, we have 2(n 1) > n, and so there must be a singleton set A = {v}. Now v is joined to all other vertices, and we conclude as above that the graph is a windmill. (b) Let G be regular with valency k. Then the adjacency matrix A satisfies A 2 = ki + (J I) = (k 1)I + J, where J is the all-1 matrix (by Question 12). Now the all-1 vector j is an eigenvector with eigenvalue k. Since A is symmetric, any other eigenvector w is orthogonal to j, whence Jw = 0, and so A 2 w = (k 1)w. So the other eigenvalues of A are ± k 1. Let their multiplicities be f and g. The sum of the eigenvalues is the trace of A, which is zero. SO k + ( f g) k 1 = 0. Thus k 1 is rational, and hence integral: that is, k = u for some positive integer u, and we have f g = (u 2 + 1)/u. 8

9 This implies that u divides 1; so u = 1, k = 2, n = 3, and the graph is a windmill with one sail. 14 The Trackwords puzzle in the Radio Times consists of nine letters arranged in a 3 3 array. It is possible to form an English word from all nine letters, where consecutive letters are adjacent horizontally, vertically or diagonally. Consider the problem of setting the puzzle; more specifically, of deciding in how many ways a given word (with all its letters distinct) can be written into the array. (a) Formulate the problem in graph-theoretic terminology. (b) (COMPUTER PROJECT.) In how many ways can it be done? The question asks for the number of Hamiltonian paths in a 9-vertex graph. This number can be determined by trying all permutations of the order of the nine vertices to see which are Hamiltonian paths, which can be calculated easily (using the algorithm of (3.12.4) to generate the permutations). 15 The following algorithm, due to Peter Johnson, makes the proof of Ore s theorem (11.5.1) constructive. Verify the details. Let G satisfy the hypotheses of Ore s theorem. Choose any circuit C on the vertex set of G. If E(C) E(G), we are done. Otherwise, let d = E(C)\E(G) ; we find a new circuit C with E(C ) \ E(G) < d. A finite number of such steps finds the required Hamiltonian circuit. Take any edge e E(C) \ E(G). The graph G with edge set E(G) E(C) \ {e} satisfies Ore s condition and has a Hamiltonian path E(C) \ {e}. Now proceed as in the proof of (11.5.1) to find a Hamiltonian circuit C in G. 15. Follow the details as given. The Hamiltonian circuit obtained in G does not contain the edge e, so it has fewer edges not in G than C. After a finite number of steps, no edges not in G are used. 9

Approximation Algorithms

Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

Euler Paths and Euler Circuits

Euler Paths and Euler Circuits An Euler path is a path that uses every edge of a graph exactly once. An Euler circuit is a circuit that uses every edge of a graph exactly once. An Euler path starts and

V. Adamchik 1 Graph Theory Victor Adamchik Fall of 2005 Plan 1. Basic Vocabulary 2. Regular graph 3. Connectivity 4. Representing Graphs Introduction A.Aho and J.Ulman acknowledge that Fundamentally, computer

136 CHAPTER 4. INDUCTION, GRAPHS AND TREES

136 TER 4. INDUCTION, GRHS ND TREES 4.3 Graphs In this chapter we introduce a fundamental structural idea of discrete mathematics, that of a graph. Many situations in the applications of discrete mathematics

DATA ANALYSIS II. Matrix Algorithms

DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where

Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

Graphs without proper subgraphs of minimum degree 3 and short cycles

Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract

8. Matchings and Factors

8. Matchings and Factors Consider the formation of an executive council by the parliament committee. Each committee needs to designate one of its members as an official representative to sit on the council,

IE 680 Special Topics in Production Systems: Networks, Routing and Logistics*

IE 680 Special Topics in Production Systems: Networks, Routing and Logistics* Rakesh Nagi Department of Industrial Engineering University at Buffalo (SUNY) *Lecture notes from Network Flows by Ahuja, Magnanti

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 13 Overview Graphs and Graph

Chapter 6: Graph Theory

Chapter 6: Graph Theory Graph theory deals with routing and network problems and if it is possible to find a best route, whether that means the least expensive, least amount of time or the least distance.

Social Media Mining. Graph Essentials

Graph Essentials Graph Basics Measures Graph and Essentials Metrics 2 2 Nodes and Edges A network is a graph nodes, actors, or vertices (plural of vertex) Connections, edges or ties Edge Node Measures

On end degrees and infinite cycles in locally finite graphs

On end degrees and infinite cycles in locally finite graphs Henning Bruhn Maya Stein Abstract We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel

Graphical degree sequences and realizations

swap Graphical and realizations Péter L. Erdös Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences MAPCON 12 MPIPKS - Dresden, May 15, 2012 swap Graphical and realizations Péter L. Erdös

3. Linear Programming and Polyhedral Combinatorics

Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

Mathematical Induction

Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,

Baltic Way 1995. Västerås (Sweden), November 12, 1995. Problems and solutions

Baltic Way 995 Västerås (Sweden), November, 995 Problems and solutions. Find all triples (x, y, z) of positive integers satisfying the system of equations { x = (y + z) x 6 = y 6 + z 6 + 3(y + z ). Solution.

SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

Labeling outerplanar graphs with maximum degree three

Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

Analysis of Algorithms, I

Analysis of Algorithms, I CSOR W4231.002 Eleni Drinea Computer Science Department Columbia University Thursday, February 26, 2015 Outline 1 Recap 2 Representing graphs 3 Breadth-first search (BFS) 4 Applications

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various

arxiv:1203.1525v1 [math.co] 7 Mar 2012

Constructing subset partition graphs with strong adjacency and end-point count properties Nicolai Hähnle haehnle@math.tu-berlin.de arxiv:1203.1525v1 [math.co] 7 Mar 2012 March 8, 2012 Abstract Kim defined

Codes for Network Switches

Codes for Network Switches Zhiying Wang, Omer Shaked, Yuval Cassuto, and Jehoshua Bruck Electrical Engineering Department, California Institute of Technology, Pasadena, CA 91125, USA Electrical Engineering

Cycles in a Graph Whose Lengths Differ by One or Two

Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDE-BERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS

Graph Theory Lecture 3: Sum of Degrees Formulas, Planar Graphs, and Euler s Theorem Spring 2014 Morgan Schreffler Office: POT 902

Graph Theory Lecture 3: Sum of Degrees Formulas, Planar Graphs, and Euler s Theorem Spring 2014 Morgan Schreffler Office: POT 902 http://www.ms.uky.edu/~mschreffler Different Graphs, Similar Properties

Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.

Extremal Wiener Index of Trees with All Degrees Odd

MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 70 (2013) 287-292 ISSN 0340-6253 Extremal Wiener Index of Trees with All Degrees Odd Hong Lin School of

Single machine parallel batch scheduling with unbounded capacity

Workshop on Combinatorics and Graph Theory 21th, April, 2006 Nankai University Single machine parallel batch scheduling with unbounded capacity Yuan Jinjiang Department of mathematics, Zhengzhou University

On the k-path cover problem for cacti

On the k-path cover problem for cacti Zemin Jin and Xueliang Li Center for Combinatorics and LPMC Nankai University Tianjin 300071, P.R. China zeminjin@eyou.com, x.li@eyou.com Abstract In this paper we

Applied Algorithm Design Lecture 5

Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design

A 2-factor in which each cycle has long length in claw-free graphs

A -factor in which each cycle has long length in claw-free graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science

Mathematics for Algorithm and System Analysis

Mathematics for Algorithm and System Analysis for students of computer and computational science Edward A. Bender S. Gill Williamson c Edward A. Bender & S. Gill Williamson 2005. All rights reserved. Preface

Network (Tree) Topology Inference Based on Prüfer Sequence

Network (Tree) Topology Inference Based on Prüfer Sequence C. Vanniarajan and Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology Madras Chennai 600036 vanniarajanc@hcl.in,

AN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC

AN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC W. T. TUTTE. Introduction. In a recent series of papers [l-4] on graphs and matroids I used definitions equivalent to the following.

Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

Outline. NP-completeness. When is a problem easy? When is a problem hard? Today. Euler Circuits

Outline NP-completeness Examples of Easy vs. Hard problems Euler circuit vs. Hamiltonian circuit Shortest Path vs. Longest Path 2-pairs sum vs. general Subset Sum Reducing one problem to another Clique

SEMITOTAL AND TOTAL BLOCK-CUTVERTEX GRAPH

CHAPTER 3 SEMITOTAL AND TOTAL BLOCK-CUTVERTEX GRAPH ABSTRACT This chapter begins with the notion of block distances in graphs. Using block distance we defined the central tendencies of a block, like B-radius

Tree-representation of set families and applications to combinatorial decompositions

Tree-representation of set families and applications to combinatorial decompositions Binh-Minh Bui-Xuan a, Michel Habib b Michaël Rao c a Department of Informatics, University of Bergen, Norway. buixuan@ii.uib.no

Linear Codes. Chapter 3. 3.1 Basics

Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length

Sum of Degrees of Vertices Theorem

Sum of Degrees of Vertices Theorem Theorem (Sum of Degrees of Vertices Theorem) Suppose a graph has n vertices with degrees d 1, d 2, d 3,...,d n. Add together all degrees to get a new number d 1 + d 2

Answer: (a) Since we cannot repeat men on the committee, and the order we select them in does not matter, ( )

1. (Chapter 1 supplementary, problem 7): There are 12 men at a dance. (a) In how many ways can eight of them be selected to form a cleanup crew? (b) How many ways are there to pair off eight women at the

8.1 Min Degree Spanning Tree

CS880: Approximations Algorithms Scribe: Siddharth Barman Lecturer: Shuchi Chawla Topic: Min Degree Spanning Tree Date: 02/15/07 In this lecture we give a local search based algorithm for the Min Degree

INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

SPANNING CACTI FOR STRUCTURALLY CONTROLLABLE NETWORKS NGO THI TU ANH NATIONAL UNIVERSITY OF SINGAPORE

SPANNING CACTI FOR STRUCTURALLY CONTROLLABLE NETWORKS NGO THI TU ANH NATIONAL UNIVERSITY OF SINGAPORE 2012 SPANNING CACTI FOR STRUCTURALLY CONTROLLABLE NETWORKS NGO THI TU ANH (M.Sc., SFU, Russia) A THESIS

6.852: Distributed Algorithms Fall, 2009. Class 2

.8: Distributed Algorithms Fall, 009 Class Today s plan Leader election in a synchronous ring: Lower bound for comparison-based algorithms. Basic computation in general synchronous networks: Leader election

Classification of Cartan matrices

Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms

SHORT CYCLE COVERS OF GRAPHS WITH MINIMUM DEGREE THREE

SHOT YLE OVES OF PHS WITH MINIMUM DEEE THEE TOMÁŠ KISE, DNIEL KÁL, END LIDIKÝ, PVEL NEJEDLÝ OET ŠÁML, ND bstract. The Shortest ycle over onjecture of lon and Tarsi asserts that the edges of every bridgeless

Max Flow, Min Cut, and Matchings (Solution)

Max Flow, Min Cut, and Matchings (Solution) 1. The figure below shows a flow network on which an s-t flow is shown. The capacity of each edge appears as a label next to the edge, and the numbers in boxes

Cycle transversals in bounded degree graphs

Electronic Notes in Discrete Mathematics 35 (2009) 189 195 www.elsevier.com/locate/endm Cycle transversals in bounded degree graphs M. Groshaus a,2,3 P. Hell b,3 S. Klein c,1,3 L. T. Nogueira d,1,3 F.

Computer Science Department. Technion - IIT, Haifa, Israel. Itai and Rodeh [IR] have proved that for any 2-connected graph G and any vertex s G there

- 1 - THREE TREE-PATHS Avram Zehavi Alon Itai Computer Science Department Technion - IIT, Haifa, Israel Abstract Itai and Rodeh [IR] have proved that for any 2-connected graph G and any vertex s G there

The positive minimum degree game on sparse graphs

The positive minimum degree game on sparse graphs József Balogh Department of Mathematical Sciences University of Illinois, USA jobal@math.uiuc.edu András Pluhár Department of Computer Science University

A Turán Type Problem Concerning the Powers of the Degrees of a Graph

A Turán Type Problem Concerning the Powers of the Degrees of a Graph Yair Caro and Raphael Yuster Department of Mathematics University of Haifa-ORANIM, Tivon 36006, Israel. AMS Subject Classification:

Stationary random graphs on Z with prescribed iid degrees and finite mean connections

Stationary random graphs on Z with prescribed iid degrees and finite mean connections Maria Deijfen Johan Jonasson February 2006 Abstract Let F be a probability distribution with support on the non-negative

USE OF EIGENVALUES AND EIGENVECTORS TO ANALYZE BIPARTIVITY OF NETWORK GRAPHS

USE OF EIGENVALUES AND EIGENVECTORS TO ANALYZE BIPARTIVITY OF NETWORK GRAPHS Natarajan Meghanathan Jackson State University, 1400 Lynch St, Jackson, MS, USA natarajan.meghanathan@jsums.edu ABSTRACT This

Cycles and clique-minors in expanders

Cycles and clique-minors in expanders Benny Sudakov UCLA and Princeton University Expanders Definition: The vertex boundary of a subset X of a graph G: X = { all vertices in G\X with at least one neighbor

A permutation can also be represented by describing its cycles. What do you suppose is meant by this?

Shuffling, Cycles, and Matrices Warm up problem. Eight people stand in a line. From left to right their positions are numbered,,,... 8. The eight people then change places according to THE RULE which directs

A Study of Sufficient Conditions for Hamiltonian Cycles

DeLeon 1 A Study of Sufficient Conditions for Hamiltonian Cycles Melissa DeLeon Department of Mathematics and Computer Science Seton Hall University South Orange, New Jersey 07079, U.S.A. ABSTRACT A graph

The Independence Number in Graphs of Maximum Degree Three

The Independence Number in Graphs of Maximum Degree Three Jochen Harant 1 Michael A. Henning 2 Dieter Rautenbach 1 and Ingo Schiermeyer 3 1 Institut für Mathematik, TU Ilmenau, Postfach 100565, D-98684

Mining Social-Network Graphs

342 Chapter 10 Mining Social-Network Graphs There is much information to be gained by analyzing the large-scale data that is derived from social networks. The best-known example of a social network is

Tiers, Preference Similarity, and the Limits on Stable Partners

Tiers, Preference Similarity, and the Limits on Stable Partners KANDORI, Michihiro, KOJIMA, Fuhito, and YASUDA, Yosuke February 7, 2010 Preliminary and incomplete. Do not circulate. Abstract We consider

P. Jeyanthi and N. Angel Benseera

Opuscula Math. 34, no. 1 (014), 115 1 http://dx.doi.org/10.7494/opmath.014.34.1.115 Opuscula Mathematica A TOTALLY MAGIC CORDIAL LABELING OF ONE-POINT UNION OF n COPIES OF A GRAPH P. Jeyanthi and N. Angel

Simple Graphs Degrees, Isomorphism, Paths

Mathematics for Computer Science MIT 6.042J/18.062J Simple Graphs Degrees, Isomorphism, Types of Graphs Simple Graph this week Multi-Graph Directed Graph next week Albert R Meyer, March 10, 2010 lec 6W.1

Small Maximal Independent Sets and Faster Exact Graph Coloring

Small Maximal Independent Sets and Faster Exact Graph Coloring David Eppstein Univ. of California, Irvine Dept. of Information and Computer Science The Exact Graph Coloring Problem: Given an undirected

Offline sorting buffers on Line

Offline sorting buffers on Line Rohit Khandekar 1 and Vinayaka Pandit 2 1 University of Waterloo, ON, Canada. email: rkhandekar@gmail.com 2 IBM India Research Lab, New Delhi. email: pvinayak@in.ibm.com

Solutions to Homework 6

Solutions to Homework 6 Debasish Das EECS Department, Northwestern University ddas@northwestern.edu 1 Problem 5.24 We want to find light spanning trees with certain special properties. Given is one example

UPPER BOUNDS ON THE L(2, 1)-LABELING NUMBER OF GRAPHS WITH MAXIMUM DEGREE

UPPER BOUNDS ON THE L(2, 1)-LABELING NUMBER OF GRAPHS WITH MAXIMUM DEGREE ANDREW LUM ADVISOR: DAVID GUICHARD ABSTRACT. L(2,1)-labeling was first defined by Jerrold Griggs [Gr, 1992] as a way to use graphs

Why? A central concept in Computer Science. Algorithms are ubiquitous.

Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online

Tenacity and rupture degree of permutation graphs of complete bipartite graphs

Tenacity and rupture degree of permutation graphs of complete bipartite graphs Fengwei Li, Qingfang Ye and Xueliang Li Department of mathematics, Shaoxing University, Shaoxing Zhejiang 312000, P.R. China

The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

The Clar Structure of Fullerenes

The Clar Structure of Fullerenes Liz Hartung Massachusetts College of Liberal Arts June 12, 2013 Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 1 / 25

Finding and counting given length cycles

Finding and counting given length cycles Noga Alon Raphael Yuster Uri Zwick Abstract We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected

Optimal Index Codes for a Class of Multicast Networks with Receiver Side Information

Optimal Index Codes for a Class of Multicast Networks with Receiver Side Information Lawrence Ong School of Electrical Engineering and Computer Science, The University of Newcastle, Australia Email: lawrence.ong@cantab.net

Fundamentals of Media Theory

Fundamentals of Media Theory ergei Ovchinnikov Mathematics Department an Francisco tate University an Francisco, CA 94132 sergei@sfsu.edu Abstract Media theory is a new branch of discrete applied mathematics

Regular Augmentations of Planar Graphs with Low Degree

Regular Augmentations of Planar Graphs with Low Degree Bachelor Thesis of Huyen Chau Nguyen At the Department of Informatics Institute of Theoretical Computer Science Reviewers: Advisors: Prof. Dr. Dorothea

THE PROBLEM WORMS (1) WORMS (2) THE PROBLEM OF WORM PROPAGATION/PREVENTION THE MINIMUM VERTEX COVER PROBLEM

1 THE PROBLEM OF WORM PROPAGATION/PREVENTION I.E. THE MINIMUM VERTEX COVER PROBLEM Prof. Tiziana Calamoneri Network Algorithms A.y. 2014/15 2 THE PROBLEM WORMS (1)! A computer worm is a standalone malware

THE COLLEGES OF OXFORD UNIVERSITY MATHEMATICS, JOINT SCHOOLS AND COMPUTER SCIENCE. Sample Solutions for Specimen Test 2

THE COLLEGES OF OXFORD UNIVERSITY MATHEMATICS, JOINT SCHOOLS AND COMPUTER SCIENCE Sample Solutions for Specimen Test. A. The vector from P (, ) to Q (8, ) is PQ =(6, 6). If the point R lies on PQ such

COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS

COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS Alexander Burstein Department of Mathematics Howard University Washington, DC 259, USA aburstein@howard.edu Sergey Kitaev Mathematics

CSC2420 Fall 2012: Algorithm Design, Analysis and Theory

CSC2420 Fall 2012: Algorithm Design, Analysis and Theory Allan Borodin November 15, 2012; Lecture 10 1 / 27 Randomized online bipartite matching and the adwords problem. We briefly return to online algorithms

Discuss the size of the instance for the minimum spanning tree problem.

3.1 Algorithm complexity The algorithms A, B are given. The former has complexity O(n 2 ), the latter O(2 n ), where n is the size of the instance. Let n A 0 be the size of the largest instance that can

Near Optimal Solutions

Near Optimal Solutions Many important optimization problems are lacking efficient solutions. NP-Complete problems unlikely to have polynomial time solutions. Good heuristics important for such problems.

THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE. Alexander Barvinok

THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE Alexer Barvinok Papers are available at http://www.math.lsa.umich.edu/ barvinok/papers.html This is a joint work with J.A. Hartigan

The Open University s repository of research publications and other research outputs

Open Research Online The Open University s repository of research publications and other research outputs The degree-diameter problem for circulant graphs of degree 8 and 9 Journal Article How to cite:

Design of LDPC codes

Design of LDPC codes Codes from finite geometries Random codes: Determine the connections of the bipartite Tanner graph by using a (pseudo)random algorithm observing the degree distribution of the code

CS311H. Prof: Peter Stone. Department of Computer Science The University of Texas at Austin

CS311H Prof: Department of Computer Science The University of Texas at Austin Good Morning, Colleagues Good Morning, Colleagues Are there any questions? Logistics Class survey Logistics Class survey Homework

On the crossing number of K m,n

On the crossing number of K m,n Nagi H. Nahas nnahas@acm.org Submitted: Mar 15, 001; Accepted: Aug 10, 00; Published: Aug 1, 00 MR Subject Classifications: 05C10, 05C5 Abstract The best lower bound known

Decision Mathematics D1 Advanced/Advanced Subsidiary. Tuesday 5 June 2007 Afternoon Time: 1 hour 30 minutes

Paper Reference(s) 6689/01 Edexcel GCE Decision Mathematics D1 Advanced/Advanced Subsidiary Tuesday 5 June 2007 Afternoon Time: 1 hour 30 minutes Materials required for examination Nil Items included with

On Integer Additive Set-Indexers of Graphs

On Integer Additive Set-Indexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A set-indexer of a graph G is an injective set-valued function f : V (G) 2 X such that

JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004

Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February

Large induced subgraphs with all degrees odd

Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order

WOLLONGONG COLLEGE AUSTRALIA. Diploma in Information Technology

First Name: Family Name: Student Number: Class/Tutorial: WOLLONGONG COLLEGE AUSTRALIA A College of the University of Wollongong Diploma in Information Technology Final Examination Spring Session 2008 WUCT121

Mean Ramsey-Turán numbers

Mean Ramsey-Turán numbers Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel Abstract A ρ-mean coloring of a graph is a coloring of the edges such that the average

Sport Timetabling. Outline DM87 SCHEDULING, TIMETABLING AND ROUTING. Outline. Lecture 15. 1. Problem Definitions

Outline DM87 SCHEDULING, TIMETABLING AND ROUTING Lecture 15 Sport Timetabling 1. Problem Definitions Marco Chiarandini DM87 Scheduling, Timetabling and Routing 2 Problems we treat: single and double round-robin

Dedekind s forgotten axiom and why we should teach it (and why we shouldn t teach mathematical induction in our calculus classes)

Dedekind s forgotten axiom and why we should teach it (and why we shouldn t teach mathematical induction in our calculus classes) by Jim Propp (UMass Lowell) March 14, 2010 1 / 29 Completeness Three common

Network File Storage with Graceful Performance Degradation

Network File Storage with Graceful Performance Degradation ANXIAO (ANDREW) JIANG California Institute of Technology and JEHOSHUA BRUCK California Institute of Technology A file storage scheme is proposed

A Sublinear Bipartiteness Tester for Bounded Degree Graphs

A Sublinear Bipartiteness Tester for Bounded Degree Graphs Oded Goldreich Dana Ron February 5, 1998 Abstract We present a sublinear-time algorithm for testing whether a bounded degree graph is bipartite

If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?

Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question

The Pointless Machine and Escape of the Clones

MATH 64091 Jenya Soprunova, KSU The Pointless Machine and Escape of the Clones The Pointless Machine that operates on ordered pairs of positive integers (a, b) has three modes: In Mode 1 the machine adds