# Diameter and Treewidth in Minor-Closed Graph Families, Revisited

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Algorithmica manuscript No. (will be inserted by the editor) Diameter and Treewidth in Minor-Closed Graph Families, Revisited Erik D. Demaine, MohammadTaghi Hajiaghayi MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar St., Cambridge, Massachusetts 02139, USA, {edemaine, The date of receipt and acceptance will be inserted by the editor Abstract Eppstein [5] characterized the minor-closed graph families for which the treewidth is bounded by a function of the diameter, which includes e.g. planar graphs. This characterization has been used as the basis for several (approximation) algorithms on such graphs (e.g., [2, 5 8]). The proof of Eppstein is complicated. In this short paper, we obtain the same characterization with a simple proof. In addition, the relation between treewidth and diameter is slightly better and explicit. 1 Introduction Eppstein [5] introduced the diameter-treewidth property for a class of graphs, which requires that the treewidth of a graph in the class is upper bounded by a function of its diameter. This notion has been used extensively in a slightly modified form called the bounded-local-treewidth property, which requires that the treewidth of any connected subgraph of a graph in the class is upper bounded by a function of its diameter. For minor-closed graph families, which is the focus of most work in this context, these properties are identical. The reason for introducing graphs of bounded local treewidth is that they have many similar properties to both planar graphs and graphs of bounded treewidth, two classes of graphs on which many problems are substantially easier. In particular, Baker s approach for polynomial-time approximation schemes (PTASs) on planar graphs [1] applies to this setting. As a result, PTASs are known for hereditary maximization problems such as maximum independent set, maximum triangle matching, maximum H-matching, maximum tile salvage, minimum vertex cover, minimum dominating set, minimum edge-dominating set, and subgraph isomorphism for a fixed pattern [2,5,8]. Graphs of bounded local treewidth also admit several efficient fixed-parameter algorithms. In particular, Frick and Grohe [6] give

2 2 Erik D. Demaine, MohammadTaghi Hajiaghayi a general framework for deciding any property expressible in first-order logic in graphs of bounded local treewidth. The foundation of these results is the following characterization by Eppstein [5] of minor-closed families with the diameter-treewidth property. An apex graph is a graph in which the removal of some vertex leaves a planar graph. Theorem 1 Let F be a minor-closed family of graphs. Then F has the diametertreewidth property if and only if F does not contain all apex graphs, i.e., F excludes some apex graph. We reprove this theorem with a much simpler proof. Similar to Eppstein s proof, we use the following theorems from Graph Minor Theory. The m m grid is the planar graph with m 2 vertices arranged on a square grid and with edges connecting horizontally and vertically adjacent vertices. Theorem 2 ([4]) For integers r and m, let G be a graph of treewidth at least m 4r2 (m+2). Then G contains either the complete graph K r or the m m grid as a minor. Theorem 3 ([9]) Every planar graph H can be obtained as a minor of the r r grid H, where r = 14 V (H) 24. For readers unfamiliar with the notions of minors and treewidth, we give the necessary definitions for background. Contracting an edge e = {u, v} in an (undirected) graph G is the operation of replacing both u and v by a single vertex w whose neighbors are all vertices that were neighbors of u or v, except u and v themselves. A graph G is a minor of a graph H if H can be obtained from a subgraph of G by contracting edges. A graph class C is minor-closed if any minor of any graph in C is also a member of C. A minor-closed graph class C is H-minor-free if H / C. Representation of a graph as a tree with a tree decomposition plays an important role in design of algorithms. A tree decomposition of a graph G = (V, E) is a pair (χ, T ) where T = (I, F ) is a tree and χ = {χ i i I} is a family of subsets of V (G) such that (1) i I χ i = V ; (2) for each edge e = {u, v} E, there exists an i I such that both u and v belong to χ i ; and (3) for all v V, the set of nodes {i I v χ i } forms a connected subtree of T. The maximum size of any χ i, minus one, is called the width of the tree decomposition. The treewidth of a graph G, denoted by tw(g), is the minimum width over all possible tree decompositions of G. 2 The Main Result First we present a property of apex-minor-free graphs. The vertices (i, j) of the m m grid with i {1, m} or j {1, m} are called boundary vertices, and the rest of the vertices in the grid are called nonboundary vertices.

3 Diameter and Treewidth in Minor-Closed Graph Families, Revisited 3 Lemma 1 Let G be an H-minor-free graph for an apex graph H, let k = 14 V (H) 22, and let m > 2k be the largest integer such that tw(g) m 4 V (H) 2 (m+2). Then G can be contracted into an augmented grid R, i.e., a (m 2k) (m 2k) grid augmented with additional edges (and no additional vertices) such that each vertex v V (R) is adjacent to less than (k + 1) 6 nonboundary vertices of the grid. Proof By Theorem 2, G contains an m m grid M as a minor. Thus there exists a sequence of edge contractions and edge/vertex deletions reducing G to M. We apply to G the edge contractions from this sequence; we ignore the edge deletions; and instead of deletion of a vertex v, we only contract v into one of its neighbors. Call the new graph G, which has the m m grid M as a subgraph and in addition V (G ) = V (M). We claim that each vertex v V (G ) is adjacent to at most k 4 vertices in the central (m 2k) (m 2k) subgrid M of M. In other words, let N be the set of neighbors of any vertex v V (G ) that are in M. We claim that N k 4. Suppose for contradiction that N > k 4. Let n x denote the number of distinct x coordinates of the vertices in N, and let n y denote the number of distinct y coordinates of the vertices in N. Thus, N n x n y. Assume by symmetry that n y n x, and therefore n y N > k 2. We define the subset N of N by removing all but one (arbitrary chosen) vertex that share a common y coordinate, for each y coordinate. Thus, all y coordinates of the vertices in N are distinct, and N = n y > k 2. We discard all but k 2 (arbitrarily chosen) vertices in N to form a slightly smaller set N. We divide these k 2 vertices into k groups each of exactly k consecutive vertices according to the order of their y coordinates. Now we construct the minor k k grid K as shown in Figure 1. Because each y coordinate is unique, we can draw long horizontal segments through every point. The k columns on the left-hand and right-hand sides of M allow us to connect these horizontal segments together into k vertex-disjoint paths, each passing through exactly k vertices of N. These paths can be connected by vertical segments within each group. This arrangement of paths has the desired k k grid K as a minor, where the vertices of the grid correspond to the vertices in N. Now, if v has been used in the contraction of a vertex v in K, we proceed as shown in Figure 2. First we lift v from the grid not removing it from the graph per se, but k Fig. 1 Construction of the minor k k grid K. marking it as outside the grid. Then we contract the remainder of v s column and the two adjacent columns (if they exist) into a single column. Similarly we contract k

4 4 Erik D. Demaine, MohammadTaghi Hajiaghayi the remainder of v s row with the adjacent rows. Thus we obtain as a minor of K a (k 2) (k 2) grid K such that vertex v is outside this grid and adjacent to all vertices of the grid. Now, by Theorem 3, because k 2 14 V (H) 24, we can consider v as the apex of H and obtain the planar part of H as a minor of K. Hence the original graph G is not H-minor-free, a contradiction. This concludes the proof of the claim that N k 4. v v v v (a) (b) Fig. 2 In the k k grid K, we (a) lift the vertex v, (b) contract the adjacent columns, and (c) contract the adjacent rows, to form a (k 2) (k 2) grid K. Vertex v is adjacent to all vertices in the grid, though the figure just shows four neighbors for visibility. (c) (d) Finally, form a new graph R by taking graph G and contracting all 2k boundary rows and 2k boundary columns into two boundary rows and two boundary columns (one on each side). The number of neighbors of each vertex of R that are not on the boundary is at most (k + 1) 2 k 4. The factor (k + 1) 2 is for the boundary vertices each of which is obtained by contraction of at most (k + 1) 2 vertices. Now we are ready to prove Theorem 1. Proof (of Theorem 1) One direction is easy. The apex graphs A i, i = 1, 2,..., obtained from the i i grid by connecting a new vertex v to all vertices of the grid have diameter two and treewidth i + 1, because the treewidth of the i i grid is i (see e.g. [3]). Thus a minor-closed family of graphs with the diametertreewidth property cannot contain all apex graphs. Next consider the other direction. Let G be a graph from a minor-closed family F of graphs excluding an apex graph H. We show that the treewidth of G is bounded above by a function of V (H) and its diameter d. Let m be the largest integer such that tw(g) m 4 V (H) 2(m+2), and let k = 14 V (H) 22. Let R be the (m 2k) (m 2k) augmented grid obtained from G by contraction, using Lemma 1. Because diameter does not increase by contraction, the diameter of R is at most d. In addition, one can easily observe that the number of vertices of distance at most i from any vertex in R is at most 4r + 4r(k + 1) 6 + 4r(k + 1) r(k + 1) 6i 4r(k + 1) 6(i+1), where r = m 2k. Because the diameter is at most d, we have 4r(k + 1) 6(d+1) r 2, i.e., m 2k + 4(k + 1) 6(d+1). Thus the treewidth of G is at most (4(k + 1) 6(d+1) + 2k + 1) 4 V (H) 2 (4(k+1) 6(d+1) +2k+3) = O( V (H) 6(d+1) ) O( V (H) 6d+8) = 2 O(d V (H) 6d+8 lg V (H) ), as desired.

5 Diameter and Treewidth in Minor-Closed Graph Families, Revisited 5 References 1. B. S. BAKER, Approximation algorithms for NP-complete problems on planar graphs, J. Assoc. Comput. Mach., 41 (1994), pp E. D. DEMAINE, M. HAJIAGHAYI, N. NISHIMURA, P. RAGDE, AND D. M. THI- LIKOS, Approximation algorithms for classes of graphs excluding single-crossing graphs as minors, Journal of Computer and System Sciences. To appear. 3. R. DIESTEL, Graph theory, Graduate Texts in Mathematics Vol. 173, Springer-Verlag, New York, R. DIESTEL, T. R. JENSEN, K. Y. GORBUNOV, AND C. THOMASSEN, Highly connected sets and the excluded grid theorem, J. Combin. Theory Ser. B, 75 (1999), pp D. EPPSTEIN, Diameter and treewidth in minor-closed graph families, Algorithmica, 27 (2000), pp M. FRICK AND M. GROHE, Deciding first-order properties of locally treedecomposable graphs, J. ACM, 48 (2001), pp M. GROHE, Local tree-width, excluded minors, and approximation algorithms. To appear in Combinatorica. 8. M. HAJIAGHAYI AND N. NISHIMURA, Subgraph isomorphism, log-bounded fragmentation and graphs of (locally) bounded treewidth, in The 27th International Symposium on Mathematical Foundations of Computer Science (Poland, 2002), 2002, pp N. ROBERTSON, P. SEYMOUR, AND R. THOMAS, Quickly excluding a planar graph, J. Combin. Theory Ser. B, 62 (1994), pp

### The Bidimensionality Theory and Its Algorithmic Applications. MohammadTaghi Hajiaghayi

The Bidimensionality Theory and Its Algorithmic Applications by MohammadTaghi Hajiaghayi B.S., Sharif University of Technology, 2000 M.S., University of Waterloo, 2001 Submitted to the Department of Mathematics

### 2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

### 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.,

### GRAPH THEORY LECTURE 4: TREES

GRAPH THEORY LECTURE 4: TREES Abstract. 3.1 presents some standard characterizations and properties of trees. 3.2 presents several different types of trees. 3.7 develops a counting method based on a bijection

### Bidimensionality and Graph Decompositions. Athanassios Koutsonas

Bidimensionality and Graph Decompositions Athanassios Koutsonas µ λ March 2008 2 Preface According to Isokrates, a great pedagogue of the classical era, truly educated is not he who bears knowledge and

### On the independence number of graphs with maximum degree 3

On the independence number of graphs with maximum degree 3 Iyad A. Kanj Fenghui Zhang Abstract Let G be an undirected graph with maximum degree at most 3 such that G does not contain any of the three graphs

### 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.

### 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

### Divisor graphs have arbitrary order and size

Divisor graphs have arbitrary order and size arxiv:math/0606483v1 [math.co] 20 Jun 2006 Le Anh Vinh School of Mathematics University of New South Wales Sydney 2052 Australia Abstract A divisor graph G

### A Structural View on Parameterizing Problems: Distance from Triviality

A Structural View on Parameterizing Problems: Distance from Triviality Jiong Guo, Falk Hüffner, and Rolf Niedermeier Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Sand 13, D-72076 Tübingen,

### 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

### Midterm Practice Problems

6.042/8.062J Mathematics for Computer Science October 2, 200 Tom Leighton, Marten van Dijk, and Brooke Cowan Midterm Practice Problems Problem. [0 points] In problem set you showed that the nand operator

### 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

### arxiv: v2 [math.co] 30 Nov 2015

PLANAR GRAPH IS ON FIRE PRZEMYSŁAW GORDINOWICZ arxiv:1311.1158v [math.co] 30 Nov 015 Abstract. Let G be any connected graph on n vertices, n. Let k be any positive integer. Suppose that a fire breaks out

### CS 598CSC: Combinatorial Optimization Lecture date: 2/4/2010

CS 598CSC: Combinatorial Optimization Lecture date: /4/010 Instructor: Chandra Chekuri Scribe: David Morrison Gomory-Hu Trees (The work in this section closely follows [3]) Let G = (V, E) be an undirected

### 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

### Clique coloring B 1 -EPG graphs

Clique coloring B 1 -EPG graphs Flavia Bonomo a,c, María Pía Mazzoleni b,c, and Maya Stein d a Departamento de Computación, FCEN-UBA, Buenos Aires, Argentina. b Departamento de Matemática, FCE-UNLP, La

### Fairness in Routing and Load Balancing

Fairness in Routing and Load Balancing Jon Kleinberg Yuval Rabani Éva Tardos Abstract We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria

### Connected Identifying Codes for Sensor Network Monitoring

Connected Identifying Codes for Sensor Network Monitoring Niloofar Fazlollahi, David Starobinski and Ari Trachtenberg Dept. of Electrical and Computer Engineering Boston University, Boston, MA 02215 Email:

### Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs

Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs Yong Zhang 1.2, Francis Y.L. Chin 2, and Hing-Fung Ting 2 1 College of Mathematics and Computer Science, Hebei University,

### Triangle deletion. Ernie Croot. February 3, 2010

Triangle deletion Ernie Croot February 3, 2010 1 Introduction The purpose of this note is to give an intuitive outline of the triangle deletion theorem of Ruzsa and Szemerédi, which says that if G = (V,

### 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

### Homework Exam 1, Geometric Algorithms, 2016

Homework Exam 1, Geometric Algorithms, 2016 1. (3 points) Let P be a convex polyhedron in 3-dimensional space. The boundary of P is represented as a DCEL, storing the incidence relationships between the

### When is a graph planar?

When is a graph planar? Theorem(Euler, 1758) If a plane multigraph G with k components has n vertices, e edges, and f faces, then n e + f = 1 + k. Corollary If G is a simple, planar graph with n(g) 3,

### HOLES 5.1. INTRODUCTION

HOLES 5.1. INTRODUCTION One of the major open problems in the field of art gallery theorems is to establish a theorem for polygons with holes. A polygon with holes is a polygon P enclosing several other

### Short Cycles make W-hard problems hard: FPT algorithms for W-hard Problems in Graphs with no short Cycles

Short Cycles make W-hard problems hard: FPT algorithms for W-hard Problems in Graphs with no short Cycles Venkatesh Raman and Saket Saurabh The Institute of Mathematical Sciences, Chennai 600 113. {vraman

### COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

### CMPSCI611: Approximating MAX-CUT Lecture 20

CMPSCI611: Approximating MAX-CUT Lecture 20 For the next two lectures we ll be seeing examples of approximation algorithms for interesting NP-hard problems. Today we consider MAX-CUT, which we proved to

### Error Compensation in Leaf Power Problems

Error Compensation in Leaf Power Problems Michael Dom Jiong Guo Falk Hüffner Rolf Niedermeier April 6, 2005 Abstract The k-leaf Power recognition problem is a particular case of graph power problems: For

### 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

### 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

### A Note on Maximum Independent Sets in Rectangle Intersection Graphs

A Note on Maximum Independent Sets in Rectangle Intersection Graphs Timothy M. Chan School of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1, Canada tmchan@uwaterloo.ca September 12,

### Shortcut sets for plane Euclidean networks (Extended abstract) 1

Shortcut sets for plane Euclidean networks (Extended abstract) 1 J. Cáceres a D. Garijo b A. González b A. Márquez b M. L. Puertas a P. Ribeiro c a Departamento de Matemáticas, Universidad de Almería,

### 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

### . 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9

Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a

### 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

### Counting Falsifying Assignments of a 2-CF via Recurrence Equations

Counting Falsifying Assignments of a 2-CF via Recurrence Equations Guillermo De Ita Luna, J. Raymundo Marcial Romero, Fernando Zacarias Flores, Meliza Contreras González and Pedro Bello López Abstract

### Graph Theory Problems and Solutions

raph Theory Problems and Solutions Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles November, 005 Problems. Prove that the sum of the degrees of the vertices of any finite graph is

### 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:

### Generalized Induced Factor Problems

Egerváry Research Group on Combinatorial Optimization Technical reports TR-2002-07. Published by the Egrerváry Research Group, Pázmány P. sétány 1/C, H 1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres.

### Discrete Applied Mathematics. The firefighter problem with more than one firefighter on trees

Discrete Applied Mathematics 161 (2013) 899 908 Contents lists available at SciVerse ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam The firefighter problem with

### 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

### Approximating the Minimum Chain Completion problem

Approximating the Minimum Chain Completion problem Tomás Feder Heikki Mannila Evimaria Terzi Abstract A bipartite graph G = (U, V, E) is a chain graph [9] if there is a bijection π : {1,..., U } U such

### Degree-3 Treewidth Sparsifiers

Degree-3 Treewidth Sparsifiers Chandra Chekuri Julia Chuzhoy Abstract We study treewidth sparsifiers. Informally, given a graph G of treewidth k, a treewidth sparsifier H is a minor of G, whose treewidth

### 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

### Answers to some of the exercises.

Answers to some of the exercises. Chapter 2. Ex.2.1 (a) There are several ways to do this. Here is one possibility. The idea is to apply the k-center algorithm first to D and then for each center in D

### Lecture 7: NP-Complete Problems

IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 7: NP-Complete Problems David Mix Barrington and Alexis Maciel July 25, 2000 1. Circuit

### BOUNDARY EDGE DOMINATION IN GRAPHS

BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 0-4874, ISSN (o) 0-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 5(015), 197-04 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA

### Ramsey numbers for bipartite graphs with small bandwidth

Ramsey numbers for bipartite graphs with small bandwidth Guilherme O. Mota 1,, Gábor N. Sárközy 2,, Mathias Schacht 3,, and Anusch Taraz 4, 1 Instituto de Matemática e Estatística, Universidade de São

### What s next? Reductions other than kernelization

What s next? Reductions other than kernelization Dániel Marx Humboldt-Universität zu Berlin (with help from Fedor Fomin, Daniel Lokshtanov and Saket Saurabh) WorKer 2010: Workshop on Kernelization Nov

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### 1 Definitions. Supplementary Material for: Digraphs. Concept graphs

Supplementary Material for: van Rooij, I., Evans, P., Müller, M., Gedge, J. & Wareham, T. (2008). Identifying Sources of Intractability in Cognitive Models: An Illustration using Analogical Structure Mapping.

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### Minimum Bisection is NP-hard on Unit Disk Graphs

Minimum Bisection is NP-hard on Unit Disk Graphs Josep Díaz 1 and George B. Mertzios 2 1 Departament de Llenguatges i Sistemes Informátics, Universitat Politécnica de Catalunya, Spain. 2 School of Engineering

### Induction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition

Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing

### arxiv:1307.2187v3 [cs.ds] 25 Aug 2013

Everything you always wanted to know about the parameterized complexity of Subgraph Isomorphism (but were afraid to ask) Dániel Marx Micha l Pilipczuk arxiv:1307.2187v3 [cs.ds] 25 Aug 2013 Abstract Given

### Problem Set 7 Solutions

8 8 Introduction to Algorithms May 7, 2004 Massachusetts Institute of Technology 6.046J/18.410J Professors Erik Demaine and Shafi Goldwasser Handout 25 Problem Set 7 Solutions This problem set is due in

### Solving Geometric Problems with the Rotating Calipers *

Solving Geometric Problems with the Rotating Calipers * Godfried Toussaint School of Computer Science McGill University Montreal, Quebec, Canada ABSTRACT Shamos [1] recently showed that the diameter of

### arxiv:1409.4299v1 [cs.cg] 15 Sep 2014

Planar Embeddings with Small and Uniform Faces Giordano Da Lozzo, Vít Jelínek, Jan Kratochvíl 3, and Ignaz Rutter 3,4 arxiv:409.499v [cs.cg] 5 Sep 04 Department of Engineering, Roma Tre University, Italy

### Faster Fixed Parameter Tractable Algorithms for Finding Feedback Vertex Sets

Faster Fixed Parameter Tractable Algorithms for Finding Feedback Vertex Sets VENKATESH RAMAN, SAKET SAURABH, AND C. R. SUBRAMANIAN The Institute of Mathematical Sciences Abstract. A feedback vertex set

### A Graph-Theoretic Network Security Game

A Graph-Theoretic Network Security Game Marios Mavronicolas 1, Vicky Papadopoulou 1, Anna Philippou 1, and Paul Spirakis 2 1 Department of Computer Science, University of Cyprus, Nicosia CY-1678, Cyprus.

### 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

### Institut für Informatik Lehrstuhl Theoretische Informatik I / Komplexitätstheorie. An Iterative Compression Algorithm for Vertex Cover

Friedrich-Schiller-Universität Jena Institut für Informatik Lehrstuhl Theoretische Informatik I / Komplexitätstheorie Studienarbeit An Iterative Compression Algorithm for Vertex Cover von Thomas Peiselt

### Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs

MCS-236: Graph Theory Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010 Chapter 7: Digraphs Strong Digraphs Definitions. A digraph is an ordered pair (V, E), where V is the set

### Optimal 3D Angular Resolution for Low-Degree Graphs

Journal of Graph Algorithms and Applications http://jgaa.info/ vol. 17, no. 3, pp. 173 200 (2013) DOI: 10.7155/jgaa.00290 Optimal 3D Angular Resolution for Low-Degree Graphs David Eppstein 1 Maarten Löffler

### Special Classes of Divisor Cordial Graphs

International Mathematical Forum, Vol. 7,, no. 35, 737-749 Special Classes of Divisor Cordial Graphs R. Varatharajan Department of Mathematics, Sri S.R.N.M. College Sattur - 66 3, Tamil Nadu, India varatharajansrnm@gmail.com

### Divide-and-Conquer. Three main steps : 1. divide; 2. conquer; 3. merge.

Divide-and-Conquer Three main steps : 1. divide; 2. conquer; 3. merge. 1 Let I denote the (sub)problem instance and S be its solution. The divide-and-conquer strategy can be described as follows. Procedure

### The Firefighter Problem: A Structural Analysis

Electronic Colloquium on Computational Complexity, Report No. 162 (2013) The Firefighter Problem: A Structural Analysis Janka Chlebíková a, Morgan Chopin b,1 a University of Portsmouth, School of Computing,

### 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

### Fast Sequential Summation Algorithms Using Augmented Data Structures

Fast Sequential Summation Algorithms Using Augmented Data Structures Vadim Stadnik vadim.stadnik@gmail.com Abstract This paper provides an introduction to the design of augmented data structures that offer

### Exact Polynomial-time Algorithm for the Clique Problem and P = NP for Clique Problem

xact Polynomial-time Algorithm for the Clique Problem and P = NP for Clique Problem Kanak Chandra Bora Department of Computer Science & ngineering Royal School of ngineering & Technology, Betkuchi, Guwahati-7810,

### Generating models of a matched formula with a polynomial delay

Generating models of a matched formula with a polynomial delay Petr Savicky Institute of Computer Science, Academy of Sciences of Czech Republic, Pod Vodárenskou Věží 2, 182 07 Praha 8, Czech Republic

### 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

### Computer Algorithms. NP-Complete Problems. CISC 4080 Yanjun Li

Computer Algorithms NP-Complete Problems NP-completeness The quest for efficient algorithms is about finding clever ways to bypass the process of exhaustive search, using clues from the input in order

### 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

### Offline 1-Minesweeper is NP-complete

Offline 1-Minesweeper is NP-complete James D. Fix Brandon McPhail May 24 Abstract We use Minesweeper to illustrate NP-completeness proofs, arguments that establish the hardness of solving certain problems.

### Every tree contains a large induced subgraph with all degrees odd

Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University

### Fast fixed-parameter tractable algorithms for nontrivial generalizations of vertex cover

Discrete Applied Mathematics 152 (2005) 229 245 www.elsevier.com/locate/dam Fast fixed-parameter tractable algorithms for nontrivial generalizations of vertex cover Naomi Nishimura a,1, Prabhakar Ragde

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

### Scheduling Shop Scheduling. Tim Nieberg

Scheduling Shop Scheduling Tim Nieberg Shop models: General Introduction Remark: Consider non preemptive problems with regular objectives Notation Shop Problems: m machines, n jobs 1,..., n operations

### Shortest Inspection-Path. Queries in Simple Polygons

Shortest Inspection-Path Queries in Simple Polygons Christian Knauer, Günter Rote B 05-05 April 2005 Shortest Inspection-Path Queries in Simple Polygons Christian Knauer, Günter Rote Institut für Informatik,

### M.S. Ibrahim 1*, N. Maculan 2 and M. Minoux 3

Pesquisa Operacional (2014) 34(1): 117-124 2014 Brazilian Operations Research Society Printed version ISSN 0101-7438 / Online version ISSN 1678-5142 www.scielo.br/pope A NOTE ON THE NP-HARDNESS OF THE

### An inequality for the group chromatic number of a graph

An inequality for the group chromatic number of a graph Hong-Jian Lai 1, Xiangwen Li 2 and Gexin Yu 3 1 Department of Mathematics, West Virginia University Morgantown, WV 26505 USA 2 Department of Mathematics

### 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:

### 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

### Smooth functions statistics

Smooth functions statistics V. I. rnold To describe the topological structure of a real smooth function one associates to it the graph, formed by the topological variety, whose points are the connected

### Pigeonhole Principle Solutions

Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such

### 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 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

### 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 }

### Testing Hereditary Properties of Non-Expanding Bounded-Degree Graphs

Testing Hereditary Properties of Non-Expanding Bounded-Degree Graphs Artur Czumaj Asaf Shapira Christian Sohler Abstract We study graph properties which are testable for bounded degree graphs in time independent

### 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

### 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,

### Disjoint Compatible Geometric Matchings

Disjoint Compatible Geometric Matchings Mashhood Ishaque Diane L. Souvaine Csaba D. Tóth Abstract We prove that for every even set of n pairwise disjoint line segments in the plane in general position,

### On Tilable Orthogonal Polygons

On Tilable Orthogonal Polygons György Csizmadia Jurek Czyzowicz Leszek G asieniec Evangelos Kranakis Eduardo Rivera-Campo Jorge Urrutia Abstract We consider rectangular tilings of orthogonal polygons with

### Bounded-width QBF is PSPACE-complete

Bounded-width QBF is PSPACE-complete Albert Atserias 1 and Sergi Oliva 2 1 Universitat Politècnica de Catalunya Barcelona, Spain atserias@lsi.upc.edu 2 Universitat Politècnica de Catalunya Barcelona, Spain

### Determination of the normalization level of database schemas through equivalence classes of attributes

Computer Science Journal of Moldova, vol.17, no.2(50), 2009 Determination of the normalization level of database schemas through equivalence classes of attributes Cotelea Vitalie Abstract In this paper,

### The chromatic spectrum of mixed hypergraphs

The chromatic spectrum of mixed hypergraphs Tao Jiang, Dhruv Mubayi, Zsolt Tuza, Vitaly Voloshin, Douglas B. West March 30, 2003 Abstract A mixed hypergraph is a triple H = (X, C, D), where X is the vertex