# GRAPH THEORY LECTURE 4: TREES

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 between labeled trees and numeric strings. 3.8 showns how binary trees can be counted by the Catalan recursion. Outline 3.1 Characterizations and Properties of Trees 3.2 Rooted Trees, Ordered Trees, and Binary Trees 3.7 Counting Labeled Trees: Prüfer Encoding 3.8 Counting Binary Trees: Catalan Recursion 1

2 2 GRAPH THEORY LECTURE 4: TREES 1. Characterizations of Trees Review from 1.5 tree = connected graph with no cycles. Def 1.1. In an undirected tree, a leaf is a vertex of degree Basic Properties of Trees. Proposition 1.1. Every tree with at least one edge has at least two leaves. Proof. Let P = v 1, v 2,..., v m be a path of maximum length in a tree T. Etc. v 3 v 1 v 2 v m w v 1 v 2 v 3 w v m Figure 1.1: The two cases in the proof of Prop 1.1.

3 GRAPH THEORY LECTURE 4: TREES 3 Corollary 1.2. If the minimum degree of a graph is at least 2, then that graph must contain a cycle. Proposition 1.3. Every tree on n vertices has exactly n 1 edges. Proof. By induction using Prop 1.1. Review from 2.3 An acyclic graph is called a forest. Review from 2.4 The number of components of a graph G is denoted c(g). Corollary 1.4. A forest G on n vertices has n c(g) edges. Proof. Apply Prop 1.3 to each of the components of G. Corollary 1.5. Any graph G on n vertices has at least n c(g) edges.

4 4 GRAPH THEORY LECTURE 4: TREES Six Different Characterizations of a Tree Trees have many possible characterizations, and each contributes to the structural understanding of graphs in a different way. The following theorem establishes some of the most useful characterizations. Theorem 1.8. Let T be a graph with n vertices. Then the following statements are equivalent. (1) T is a tree. (2) T contains no cycles and has n 1 edges. (3) T is connected and has n 1 edges. (4) T is connected, and every edge is a cut-edge. (5) Any two vertices of T are connected by exactly one path. (6) T contains no cycles, and for any new edge e, the graph T + e has exactly one cycle. Proof. See text.

5 The Center of a Tree Review from 1.4 and 2.3 GRAPH THEORY LECTURE 4: TREES 5 The eccentricity of a vertex v in a graph G, denoted ecc(v), is the distance from v to a vertex farthest from v. That is, ecc(v) = max x V G {d(v, x)} A central vertex of a graph is a vertex with minimum eccentricity. The center of a graph G, denoted Z(G), is the subgraph induced on the set of central vertices of G. In an arbitrary graph G, the center Z(G) can be anything from a single vertex to all of G.

6 6 GRAPH THEORY LECTURE 4: TREES However, C. Jordan showed in 1869 that the center of a tree has only two possible cases. We begin with some preliminary results concerning the eccentricity of vertices in a tree. Lemma 1.9. Let T be a tree with at least three vertices. (a) If v is a leaf of T and w is its neighbor, then ecc(w) = ecc(v) 1 (b) If u is a central vertex of T, then deg(u) 2 Proof. (a) Since T has at least three vertices, deg(w) 2 Then there exists a vertex z v such that d(w, z) = ecc(w) v w z Figure 1.-2: But z is also a vertex farthest from v, and hence ecc(v) = d(v, z) = d(w, z) + 1 = ecc(w) + 1 (b) By Part (a), a vertex of degree 1 cannot have minimum eccentricity in tree T, and hence, cannot be a central vertex of T.

7 GRAPH THEORY LECTURE 4: TREES 7 Lemma Let v and w be two vertices in a tree T such that w is of maximum distance from v (i.e., ecc(v) = d(v, w)). Then w is a leaf. Proof. Let P be the unique v-w path in tree T. If deg(w) 2, then w would have a neighbor z whose distance from v would equal d(v, w) + 1, contradicting the premise that w is at maximum distance. T P v w z Figure 1.-3:

8 8 GRAPH THEORY LECTURE 4: TREES Lemma Let T be a tree with at least three vertices, and let T be the subtree of T obtained by deleting from T all its leaves. If v is a vertex of T, then ecc T (v) = ecc T (v) + 1 Proof. Let w be a vertex of T such that ecc T (v) = d(v, w) By Lemma 1.10, vertex w is a leaf of tree T and hence, w V T (as illustrated in Figure 1.3). It follows that the neighbor of w, say z, is a vertex of T that is farthest from v among all vertices in T, that is, ecc T (v) = d(v, z). Thus, ecc T (v) = d(v, w) = d(v, z) + 1 = ecc T (v) + 1 w z T* T v Figure 1.3:

9 GRAPH THEORY LECTURE 4: TREES 9 Proposition Let T be a tree with at least three vertices, and let T be the subtree of T obtained by deleting from T all its leaves. Then Z(T ) = Z(T ) Proof. From the two preceding lemmas, we see that deleting all the leaves decreases the eccentricity of every remaining vertex by 1. It follows that the resulting tree has the same center. Corollary 1.13 (Jordan, 1869). Let T be an n-vertex tree. Then the center Z(G) is either a single vertex or a single edge. Proof. The assertion is trivially true for n = 1 and n = 2. The result follows by induction, using Proposition 1.12.

10 10 GRAPH THEORY LECTURE 4: TREES Tree Isomorphisms and Automorphisms Example 1.1. The two graphs in Fig 1.4 have the same degree sequence, but they can be readily seen to be non-isom in several ways. For instance, the center of the left graph is a single vertex, but the center of the right graph is a single edge. Also, the two graphs have unequal diameters. Figure 1.4: Why are these trees non-isomorphic?

11 GRAPH THEORY LECTURE 4: TREES 11 Example 1.2. The graph shown in Figure 1.5 below does not have a nontrivial automorphism because the three leaves are all different distances from the center, and hence, an automorphism must map each of them to itself. Figure 1.5: A tree that has no non-trivial automorphisms. Remark 1.1. There is a linear-time algorithm for testing the isomorphism of two trees (see [AhHoUl74, p84]).

12 12 GRAPH THEORY LECTURE 4: TREES Rooted Trees 2. Rooted, Ordered, Binary Trees Def 2.1. A directed tree is a directed graph whose underlying graph is a tree. Def 2.2. A rooted tree is a tree with a designated vertex called the root. Each edge is implicitly directed away from the root. r r Figure 2.1: Two common ways of drawing a rooted tree.

13 Rooted Tree Terminology GRAPH THEORY LECTURE 4: TREES 13 Designating a root imposes a hierarchy on the vertices of a rooted tree, according to their distance from that root. Def 2.3. In a rooted tree, the depth or level of a vertex v is its distance from the root, i.e., the length of the unique path from the root to v. Thus, the root has depth 0. Def 2.4. The height of a rooted tree is the length of a longest path from the root (or the greatest depth in the tree). Def 2.5. If vertex v immediately precedes vertex w on the path from the root to w, then v is parent of w and w is child of v. Def 2.6. Vertices having the same parent are called siblings. Def 2.7. A vertex w is called a descendant of a vertex v (and v is called an ancestor of w), if v is on the unique path from the root to w. If, in addition, w v, then w is a proper descendant of v (and v is a proper ancestor of w).

14 14 GRAPH THEORY LECTURE 4: TREES Def 2.8. A leaf in a rooted tree is any vertex having no children. Def 2.9. An internal vertex in a rooted tree is any vertex that has at least one child. The root is internal, unless the tree is trivial (i.e., a single vertex). Example 2.2. The height of this tree is 3. Also, r, a, b, c, and d are the internal vertices; vertices e, f, g, h, i, and j are the leaves; vertices g, h, and i are siblings; vertex a is an ancestor of j; and j is a descendant of a. r a b c d e f g h i j Figure 2.6:

15 GRAPH THEORY LECTURE 4: TREES 15 Many applications impose an upper bound on the number of children that a given vertex can have. Def An m-ary tree (m 2) is a rooted tree in which every vertex has m or fewer children. Def A complete m-ary tree is an m-ary tree in which every internal vertex has exactly m children and all leaves have the same depth. Example 2.3. Fig 2.7 shows two ternary (3-ary) trees; the one on the left is complete; the other one is not. r r Figure 2.7: Two 3-ary trees: one complete, one incomplete.

16 16 GRAPH THEORY LECTURE 4: TREES Isomorphism of Rooted Trees Def Two rooted trees are said to be isomorphic as rooted trees if there is a graph isomorphism between them that maps root to root. Example 2.4. There are more isomorphism types of rooted trees than there are of trees. Figure 2.8: Isom trees need not be isom as rooted trees.

17 Ordered Trees GRAPH THEORY LECTURE 4: TREES 17 Def An ordered tree is a rooted tree in which the children of each vertex are assigned a fixed ordering. Def In a standard plane drawing of an ordered tree, the root is at the top, the vertices at each level are horizontally aligned, and the left-to-right order of the vertices agrees with their prescribed order. Remark 2.1. In an ordered tree, the prescribed local ordering of the children of each vertex extends to several possible global orderings of the vertices of the tree. One of them, the level order, is equivalent to reading the vertex names top-to-bottom, left-to-right in a standard plane drawing. Level order and three other global orderings, pre-order, post-order, and in-order, are explored in 3.3. Example 2.6. The ordered tree on the left stores the expression a b c, whereas the one on the right stores c a b. - - * c c * a b a b Figure 2.10: Two trees: isom as rooted, not as ordered.

18 18 GRAPH THEORY LECTURE 4: TREES Binary Trees Def A binary tree is an ordered 2-ary tree in which each child is designated either a left-child or a right-child. Figure 2.11: A binary tree of height 4. Def The left (right) subtree of a vertex v in a binary tree is the binary subtree spanning the left (right)-child of v and all of its descendants.

19 GRAPH THEORY LECTURE 4: TREES 19 The mandatory designation of left-child or right-child means that two different binary trees may be indistinguishable when regarded as ordered trees. Figure 2.12: One ordered tree yielding four binary trees. Theorem 2.1. The complete binary tree of height h has 2 h+1 1 vertices. Corollary 2.2. Every binary tree of height h has at most 2 h+1 1 vertices. Figure 2.13: Complete binary tree of ht 3 has 15 vertices.

20 20 GRAPH THEORY LECTURE 4: TREES 7. Counting Labeled Trees The number of n-vertex labeled trees is n n 2, for n 2, and is known as Cayley s Formula Figure 7.1: Two different labeled trees, isom as unlabeled. Sometimes, labeled graphs are merely graphs with labels. At other times, they are a class of graph objects whose isomorphisms must preserve the labels. Cayley s formula counts the number of classes of n-vertex trees under the equivalence relation of label-preserving isomorphism.

21 GRAPH THEORY LECTURE 4: TREES 21 SUPPLEMENT: EXAMPLES OF COUNTING TREES WITH CAYLEY S FORMULA EXAMPLE: T 3 = 3 3!2 = EXAMPLE: T 4 = 4 4!2 = =16 EXAMPLE: T 5 = 5 5!2 = =125

22 22 GRAPH THEORY LECTURE 4: TREES Prüfer Encoding Def 7.1. A Prüfer sequence of length n 2, for n 2, is any sequence of integers between 1 and n, with repetitions allowed. Table 7.1: Prüfer Encoding Algorithm: Prüfer Encoding Input: an n-vertex tree with std 1-based vertex-labels. Output: a Prüfer sequence of length n 2. Initialize T to be the given tree. For i = 1 to n 2 Let v be the 1-valent vertex with the smallest label. Let s i be the label of the only neighbor of v. T := T v. Return sequence s 1, s 2,..., s n 2.

23 Example 7.1. GRAPH THEORY LECTURE 4: TREES 23 Figure 7.2: A labeled tree, to be encoded into a Prüfer sequence S. S = (7, S = (7, 4, Figure 7.3: First two iterations of the Prüfer encoding. S = (7, 4, 4, S = (7, 4, 4, 7, S = (7, 4, 4, 7, 5) Figure 7.4: Last three iterations of the Prüfer encoding.

24 24 GRAPH THEORY LECTURE 4: TREES Notice for the above example that the degree of each vertex is one more than the number of times that its label appears in the Prüfer sequence. The next result shows this is true in general. In the proof, l(u) denotes the label on vertex u. Proposition 7.1. Let d k be the # occurrences of the label k in a Prüfer encoding sequence for a labeled tree T. Then the degree of vertex k in the tree T equals d k + 1. Proof. See text. Prüfer Decoding Table 7.2: Prüfer Decoding Algorithm: Prüfer Decoding Input: a Prüfer sequence of length n 2. Output: an n-vertex tree with std 1-based vertex-labels. Initialize list P as the Prüfer input sequence. Initialize list L as 1,..., n. Initialize forest F as n isolated vertices, labeled 1 to n. For i = 1 to n 2 Let k be the smallest # in list L that is not in list P. Let j be the first number in list P. Add an edge joining the vertices labeled k and j. Remove k from list L. Remove the first occurrence of j from list P. Add an edge joining the vertices labeled with the two numbers in list L. Return F with its vertex-labeling. remaining

25 GRAPH THEORY LECTURE 4: TREES 25 Example 7.2. Decoding for the input seq 7, 4, 4, 7, 5. L = 1, 2, 3, 4, 5, 6, 7 P = 7, 4, 4, 7, 5 L = 2, 3, 4, 5, 6, 7 P = 4, 4, 7, 5 L = 3, 4, 5, 6, 7 P = 4, 7, 5 L = 4, 5, 6, 7 P = 7, 5

26 26 GRAPH THEORY LECTURE 4: TREES L = 5, 6, 7 P = 5 L = 5, 7 Theorem 7.4 (Cayley s Tree Formula). The number of different trees on n labeled vertices is n n 2. Proof. Details in text. Remark 7.1. A slightly different view of Cayley s Tree Formula is that it gives us the number of different spanning trees of the complete graph K n. The next chapter is devoted to spanning trees.

27 GRAPH THEORY LECTURE 4: TREES Counting Binary Trees Let b n denote the number of binary trees on n vertices. Then b n = b 0 b n 1 + b 1 b n b n 1 b 0 This recurrence relation is known as the Catalan recursion, and the quantity b n is called the n th Catalan number. Example 8.1. Applying the Catalan recursion to the cases n = 2 and n = 3 yields b 2 = 2 and b 3 = 5. Figure 8.1 shows the five different binary trees on 3 vertices. Figure 8.1: The five different binary trees on 3 vertices. With the aid of generating functions, it is possible to derive the following closed formula for b n. Theorem 8.1. The number b n of different binary trees on n vertices is given by b n = 1 ( ) 2n n + 1 n

28 28 GRAPH THEORY LECTURE 4: TREES 9. Supplementary Exercises Exercise 3 What are the minimum and maximum number of vertex orbits in an n-vertex tree? Exercise 16 Exercise 17 Prove that there is no 5-vertex rigid tree. Prove that there is no 6-vertex rigid tree.

### 4 Basics of Trees. Petr Hliněný, FI MU Brno 1 FI: MA010: Trees and Forests

4 Basics of Trees Trees, actually acyclic connected simple graphs, are among the simplest graph classes. Despite their simplicity, they still have rich structure and many useful application, such as in

### Theorem A graph T is a tree if, and only if, every two distinct vertices of T are joined by a unique path.

Chapter 3 Trees Section 3. Fundamental Properties of Trees Suppose your city is planning to construct a rapid rail system. They want to construct the most economical system possible that will meet the

### Trees. Definition: A tree is a connected undirected graph with no simple circuits. Example: Which of these graphs are trees?

Section 11.1 Trees Definition: A tree is a connected undirected graph with no simple circuits. Example: Which of these graphs are trees? Solution: G 1 and G 2 are trees both are connected and have no simple

### Chapter 4. Trees. 4.1 Basics

Chapter 4 Trees 4.1 Basics A tree is a connected graph with no cycles. A forest is a collection of trees. A vertex of degree one, particularly in a tree, is called a leaf. Trees arise in a variety of applications.

### CMSC 451: Graph Properties, DFS, BFS, etc.

CMSC 451: Graph Properties, DFS, BFS, etc. Slides By: Carl Kingsford Department of Computer Science University of Maryland, College Park Based on Chapter 3 of Algorithm Design by Kleinberg & Tardos. Graphs

### Counting Spanning Trees

Counting Spanning Trees Bang Ye Wu Kun-Mao Chao Counting Spanning Trees This book provides a comprehensive introduction to the modern study of spanning trees. A spanning tree for a graph G is a subgraph

### Solutions to Exercises 8

Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 8 (1) Suppose that G is a graph in which every vertex has degree at least k, where k 1, and in which every cycle contains at least 4 vertices.

### Graph Theory Notes. Vadim Lozin. Institute of Mathematics University of Warwick

Graph Theory Notes Vadim Lozin Institute of Mathematics University of Warwick 1 Introduction A graph G = (V, E) consists of two sets V and E. The elements of V are called the vertices and the elements

### Graph Theory. Introduction. Distance in Graphs. Trees. Isabela Drămnesc UVT. Computer Science Department, West University of Timişoara, Romania

Graph Theory Introduction. Distance in Graphs. Trees Isabela Drămnesc UVT Computer Science Department, West University of Timişoara, Romania November 2016 Isabela Drămnesc UVT Graph Theory and Combinatorics

### Trees and Fundamental Circuits

Trees and Fundamental Circuits Tree A connected graph without any circuits. o must have at least one vertex. o definition implies that it must be a simple graph. o only finite trees are being considered

### GRAPH THEORY and APPLICATIONS. Trees

GRAPH THEORY and APPLICATIONS Trees Properties Tree: a connected graph with no cycle (acyclic) Forest: a graph with no cycle Paths are trees. Star: A tree consisting of one vertex adjacent to all the others.

### Outline 2.1 Graph Isomorphism 2.2 Automorphisms and Symmetry 2.3 Subgraphs, part 1

GRAPH THEORY LECTURE STRUCTURE AND REPRESENTATION PART A Abstract. Chapter focuses on the question of when two graphs are to be regarded as the same, on symmetries, and on subgraphs.. discusses the concept

### COT5405 Analysis of Algorithms Homework 3 Solutions

COT0 Analysis of Algorithms Homework 3 Solutions. Prove or give a counter example: (a) In the textbook, we have two routines for graph traversal - DFS(G) and BFS(G,s) - where G is a graph and s is any

### 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

### Unicyclic Graphs with Given Number of Cut Vertices and the Maximal Merrifield - Simmons Index

Filomat 28:3 (2014), 451 461 DOI 10.2298/FIL1403451H Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Unicyclic Graphs with Given Number

### Homework 15 Solutions

PROBLEM ONE (Trees) Homework 15 Solutions 1. Recall the definition of a tree: a tree is a connected, undirected graph which has no cycles. Which of the following definitions are equivalent to this definition

### 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.

### CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992

CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992 Original Lecture #6: 28 January 1991 Topics: Triangulating Simple Polygons

### Discrete Mathematics, Chapter 5: Induction and Recursion

Discrete Mathematics, Chapter 5: Induction and Recursion Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 5 1 / 20 Outline 1 Well-founded

### Chapter 4: Trees. 2. Theorem: Let T be a graph with n vertices. Then the following statements are equivalent:

9 Properties of Trees. Definitions: Chapter 4: Trees forest - a graph that contains no cycles tree - a connected forest. Theorem: Let T be a graph with n vertices. Then the following statements are equivalent:

### Lecture 1: Course overview, circuits, and formulas

Lecture 1: Course overview, circuits, and formulas Topics in Complexity Theory and Pseudorandomness (Spring 2013) Rutgers University Swastik Kopparty Scribes: John Kim, Ben Lund 1 Course Information Swastik

### (a) (b) (c) Figure 1 : Graphs, multigraphs and digraphs. If the vertices of the leftmost figure are labelled {1, 2, 3, 4} in clockwise order from

4 Graph Theory Throughout these notes, a graph G is a pair (V, E) where V is a set and E is a set of unordered pairs of elements of V. The elements of V are called vertices and the elements of E are called

### Chapter 2 Paths and Searching

Chapter 2 Paths and Searching Section 2.1 Distance Almost every day you face a problem: You must leave your home and go to school. If you are like me, you are usually a little late, so you want to take

### Chapter There are non-isomorphic rooted trees with four vertices. Ans: 4.

Use the following to answer questions 1-26: In the questions below fill in the blanks. Chapter 10 1. If T is a tree with 999 vertices, then T has edges. 998. 2. There are non-isomorphic trees with four

### Forests and Trees: A forest is a graph with no cycles, a tree is a connected forest.

2 Trees What is a tree? Forests and Trees: A forest is a graph with no cycles, a tree is a connected forest. Theorem 2.1 If G is a forest, then comp(g) = V (G) E(G). Proof: We proceed by induction on E(G).

### Sample Problems in Discrete Mathematics

Sample Problems in Discrete Mathematics This handout lists some sample problems that you should be able to solve as a pre-requisite to Computer Algorithms Try to solve all of them You should also read

### 10. Graph Matrices Incidence Matrix

10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a

### 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 }

### COLORED GRAPHS AND THEIR PROPERTIES

COLORED GRAPHS AND THEIR PROPERTIES BEN STEVENS 1. Introduction This paper is concerned with the upper bound on the chromatic number for graphs of maximum vertex degree under three different sets of coloring

### 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

### Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS

Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS 1.1.25 Prove that the Petersen graph has no cycle of length 7. Solution: There are 10 vertices in the Petersen graph G. Assume there is a cycle C

### 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.,

### Mathematical Induction. Lecture 10-11

Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach

### Planar Tree Transformation: Results and Counterexample

Planar Tree Transformation: Results and Counterexample Selim G Akl, Kamrul Islam, and Henk Meijer School of Computing, Queen s University Kingston, Ontario, Canada K7L 3N6 Abstract We consider the problem

### 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 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:

### 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

### A tree can be defined in a variety of ways as is shown in the following theorem: 2. There exists a unique path between every two vertices of G.

7 Basic Properties 24 TREES 7 Basic Properties Definition 7.1: A connected graph G is called a tree if the removal of any of its edges makes G disconnected. A tree can be defined in a variety of ways as

### Two General Methods to Reduce Delay and Change of Enumeration Algorithms

ISSN 1346-5597 NII Technical Report Two General Methods to Reduce Delay and Change of Enumeration Algorithms Takeaki Uno NII-2003-004E Apr.2003 Two General Methods to Reduce Delay and Change of Enumeration

### 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

### Characterizations of Arboricity of Graphs

Characterizations of Arboricity of Graphs Ruth Haas Smith College Northampton, MA USA Abstract The aim of this paper is to give several characterizations for the following two classes of graphs: (i) graphs

### Lecture Notes on Spanning Trees

Lecture Notes on Spanning Trees 15-122: Principles of Imperative Computation Frank Pfenning Lecture 26 April 26, 2011 1 Introduction In this lecture we introduce graphs. Graphs provide a uniform model

### Lecture 4: The Chromatic Number

Introduction to Graph Theory Instructor: Padraic Bartlett Lecture 4: The Chromatic Number Week 1 Mathcamp 2011 In our discussion of bipartite graphs, we mentioned that one way to classify bipartite graphs

### Growth Series for Rooted Trees Katholieke Universiteit Leuven 2 June 2014

Growth Series for Rooted Trees Katholieke Universiteit Leuven 2 June 2014 recent research results by Ian Dalton and Eric Freden Motivation 1: Growth series for groups are usually computed by finding normal

### 1 Basic Definitions and Concepts in Graph Theory

CME 305: Discrete Mathematics and Algorithms 1 Basic Definitions and Concepts in Graph Theory A graph G(V, E) is a set V of vertices and a set E of edges. In an undirected graph, an edge is an unordered

### Combinatorics: The Fine Art of Counting

Combinatorics: The Fine Art of Counting Week 9 Lecture Notes Graph Theory For completeness I have included the definitions from last week s lecture which we will be using in today s lecture along with

### Principle of (Weak) Mathematical Induction. P(1) ( n 1)(P(n) P(n + 1)) ( n 1)(P(n))

Outline We will cover (over the next few weeks) Mathematical Induction (or Weak Induction) Strong (Mathematical) Induction Constructive Induction Structural Induction Principle of (Weak) Mathematical Induction

### Binary Search Trees CMPSC 122

Binary Search Trees CMPSC 122 Note: This notes packet has significant overlap with the first set of trees notes I do in CMPSC 360, but goes into much greater depth on turning BSTs into pseudocode than

### Analysis of Algorithms I: Binary Search Trees

Analysis of Algorithms I: Binary Search Trees Xi Chen Columbia University Hash table: A data structure that maintains a subset of keys from a universe set U = {0, 1,..., p 1} and supports all three dictionary

### 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

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

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

### On Some Vertex Degree Based Graph Invariants

MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 65 (20) 723-730 ISSN 0340-6253 On Some Vertex Degree Based Graph Invariants Batmend Horoldagva a and Ivan

### Prime Numbers. Chapter Primes and Composites

Chapter 2 Prime Numbers The term factoring or factorization refers to the process of expressing an integer as the product of two or more integers in a nontrivial way, e.g., 42 = 6 7. Prime numbers are

### Trees and structural induction

Trees and structural induction Margaret M. Fleck 25 October 2010 These notes cover trees, tree induction, and structural induction. (Sections 10.1, 4.3 of Rosen.) 1 Why trees? Computer scientists are obsessed

### Basic Notions on Graphs. Planar Graphs and Vertex Colourings. Joe Ryan. Presented by

Basic Notions on Graphs Planar Graphs and Vertex Colourings Presented by Joe Ryan School of Electrical Engineering and Computer Science University of Newcastle, Australia Planar graphs Graphs may be drawn

### DO NOT RE-DISTRIBUTE THIS SOLUTION FILE

Professor Kindred Math 04 Graph Theory Homework 7 Solutions April 3, 03 Introduction to Graph Theory, West Section 5. 0, variation of 5, 39 Section 5. 9 Section 5.3 3, 8, 3 Section 7. Problems you should

### 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

### 2. (a) Explain the strassen s matrix multiplication. (b) Write deletion algorithm, of Binary search tree. [8+8]

Code No: R05220502 Set No. 1 1. (a) Describe the performance analysis in detail. (b) Show that f 1 (n)+f 2 (n) = 0(max(g 1 (n), g 2 (n)) where f 1 (n) = 0(g 1 (n)) and f 2 (n) = 0(g 2 (n)). [8+8] 2. (a)

### 7. Colourings. 7.1 Vertex colouring

7. Colourings Colouring is one of the important branches of graph theory and has attracted the attention of almost all graph theorists, mainly because of the four colour theorem, the details of which can

### On Total Domination in Graphs

University of Houston - Downtown Senior Project - Fall 2012 On Total Domination in Graphs Author: David Amos Advisor: Dr. Ermelinda DeLaViña Senior Project Committee: Dr. Sergiy Koshkin Dr. Ryan Pepper

### 14. Trees and Their Properties

14. Trees and Their Properties This is the first of a few articles in which we shall study a special and important family of graphs, called trees, discuss their properties and introduce some of their applications.

### 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

### 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

### Planarity Planarity

Planarity 8.1 71 Planarity Up until now, graphs have been completely abstract. In Topological Graph Theory, it matters how the graphs are drawn. Do the edges cross? Are there knots in the graph structure?

### 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

### Any two nodes which are connected by an edge in a graph are called adjacent node.

. iscuss following. Graph graph G consist of a non empty set V called the set of nodes (points, vertices) of the graph, a set which is the set of edges and a mapping from the set of edges to a set of pairs

### MIX-DECOMPOSITON OF THE COMPLETE GRAPH INTO DIRECTED FACTORS OF DIAMETER 2 AND UNDIRECTED FACTORS OF DIAMETER 3. University of Split, Croatia

GLASNIK MATEMATIČKI Vol. 38(58)(2003), 2 232 MIX-DECOMPOSITON OF THE COMPLETE GRAPH INTO DIRECTED FACTORS OF DIAMETER 2 AND UNDIRECTED FACTORS OF DIAMETER 3 Damir Vukičević University of Split, Croatia

### Tree isomorphism. Alexander Smal. Joint Advanced Student School St.Petersburg State University of Information Technologies, Mechanics and Optics

Tree isomorphism Alexander Smal St.Petersburg State University of Information Technologies, Mechanics and Optics Joint Advanced Student School 2008 1 / 22 Motivation In some applications the chemical structures

### CSL851: Algorithmic Graph Theory Semester I Lecture 1: July 24

CSL851: Algorithmic Graph Theory Semester I 2013-2014 Lecture 1: July 24 Lecturer: Naveen Garg Scribes: Suyash Roongta Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have

### Full and Complete Binary Trees

Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full

### A simple existence criterion for normal spanning trees in infinite graphs

1 A simple existence criterion for normal spanning trees in infinite graphs Reinhard Diestel Halin proved in 1978 that there exists a normal spanning tree in every connected graph G that satisfies the

### 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

### Constant Time Generation of Free Trees

Constant Time Generation of Free Trees Michael J. Dinneen Department of Computer Science University of Auckland Class: Combinatorial Algorithms CompSci 7 Introduction This is a combinatorial algorithms

### An algorithmic classification of open surfaces

An algorithmic classification of open surfaces Sylvain Maillot January 8, 2013 Abstract We propose a formulation for the homeomorphism problem for open n-dimensional manifolds and use the Kerekjarto classification

### Math 443/543 Graph Theory Notes 4: Connector Problems

Math 443/543 Graph Theory Notes 4: Connector Problems David Glickenstein September 19, 2012 1 Trees and the Minimal Connector Problem Here is the problem: Suppose we have a collection of cities which we

### A graph is called a tree, if it is connected and has no cycles. A star is a tree with one vertex adjacent to all other vertices.

1 Trees A graph is called a tree, if it is connected and has no cycles. A star is a tree with one vertex adjacent to all other vertices. Theorem 1 For every simple graph G with n 1 vertices, the following

### Partitioning edge-coloured complete graphs into monochromatic cycles and paths

arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edge-coloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political

### Trees. Tree Definitions: Let T = (V, E, r) be a tree. The size of a tree denotes the number of nodes of the tree.

Trees After lists (including stacks and queues) and hash tables, trees represent one of the most widely used data structures. On one hand trees can be used to represent objects that are recursive in nature

### 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

### Graphs and Network Flows IE411 Lecture 1

Graphs and Network Flows IE411 Lecture 1 Dr. Ted Ralphs IE411 Lecture 1 1 References for Today s Lecture Required reading Sections 17.1, 19.1 References AMO Chapter 1 and Section 2.1 and 2.2 IE411 Lecture

### Long questions answer Advanced Mathematics for Computer Application If P= , find BT. 19. If B = 1 0, find 2B and -3B.

Unit-1: Matrix Algebra Short questions answer 1. What is Matrix? 2. Define the following terms : a) Elements matrix b) Row matrix c) Column matrix d) Diagonal matrix e) Scalar matrix f) Unit matrix OR

### On a tree graph dened by a set of cycles

Discrete Mathematics 271 (2003) 303 310 www.elsevier.com/locate/disc Note On a tree graph dened by a set of cycles Xueliang Li a,vctor Neumann-Lara b, Eduardo Rivera-Campo c;1 a Center for Combinatorics,

### On Pebbling Graphs by their Blocks

On Pebbling Graphs by their Blocks Dawn Curtis, Taylor Hines, Glenn Hurlbert, Tatiana Moyer Department of Mathematics and Statistics Arizona State University, Tempe, AZ 85287-1804 November 19, 2008 dawn.curtis@asu.edu

### Mathematical Induction

MCS-236: Graph Theory Handout #A5 San Skulrattanakulchai Gustavus Adolphus College Sep 15, 2010 Mathematical Induction The following three principles governing N are equivalent. Ordinary Induction Principle.

### root node level: internal node edge leaf node CS@VT Data Structures & Algorithms 2000-2009 McQuain

inary Trees 1 A binary tree is either empty, or it consists of a node called the root together with two binary trees called the left subtree and the right subtree of the root, which are disjoint from each

### Scheduling. Open shop, job shop, flow shop scheduling. Related problems. Open shop, job shop, flow shop scheduling

Scheduling Basic scheduling problems: open shop, job shop, flow job The disjunctive graph representation Algorithms for solving the job shop problem Computational complexity of the job shop problem Open

### 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,

### A. V. Gerbessiotis CS Spring 2014 PS 3 Mar 24, 2014 No points

A. V. Gerbessiotis CS 610-102 Spring 2014 PS 3 Mar 24, 2014 No points Problem 1. Suppose that we insert n keys into a hash table of size m using open addressing and uniform hashing. Let p(n, m) be the

### 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

### Turing Degrees and Definability of the Jump. Theodore A. Slaman. University of California, Berkeley. CJuly, 2005

Turing Degrees and Definability of the Jump Theodore A. Slaman University of California, Berkeley CJuly, 2005 Outline Lecture 1 Forcing in arithmetic Coding and decoding theorems Automorphisms of countable

### Lattice Point Geometry: Pick s Theorem and Minkowski s Theorem. Senior Exercise in Mathematics. Jennifer Garbett Kenyon College

Lattice Point Geometry: Pick s Theorem and Minkowski s Theorem Senior Exercise in Mathematics Jennifer Garbett Kenyon College November 18, 010 Contents 1 Introduction 1 Primitive Lattice Triangles 5.1

### On Graph Powers for Leaf-Labeled Trees

Journal of Algorithms 42, 69 108 (2002) doi:10.1006/jagm.2001.1195, available online at http://www.idealibrary.com on On Graph Powers for Leaf-Labeled Trees Naomi Nishimura 1 and Prabhakar Ragde 2 Department

### 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

### / Approximation Algorithms Lecturer: Michael Dinitz Topic: Steiner Tree and TSP Date: 01/29/15 Scribe: Katie Henry

600.469 / 600.669 Approximation Algorithms Lecturer: Michael Dinitz Topic: Steiner Tree and TSP Date: 01/29/15 Scribe: Katie Henry 2.1 Steiner Tree Definition 2.1.1 In the Steiner Tree problem the input

### Chapter 6 Planarity. Section 6.1 Euler s Formula

Chapter 6 Planarity Section 6.1 Euler s Formula In Chapter 1 we introduced the puzzle of the three houses and the three utilities. The problem was to determine if we could connect each of the three utilities

### 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

### 1 Plane and Planar Graphs. Definition 1 A graph G(V,E) is called plane if

Plane and Planar Graphs Definition A graph G(V,E) is called plane if V is a set of points in the plane; E is a set of curves in the plane such that. every curve contains at most two vertices and these

### Strong Induction. Catalan Numbers.

.. The game starts with a stack of n coins. In each move, you divide one stack into two nonempty stacks. + + + + + + + + + + If the new stacks have height a and b, then you score ab points for the move.