The Grammar According to West By D.B. West. 5. Expressions as units. There exists i < j with x i = x j. ( Double-Duty Definition of i, not OK ) There.

Slides:

Advertisements

Similar presentations

THE LIST COLORING CONJECTURE nicla bernasconi. topics Introduction – The LCC Kernels and choosability Proof of the bipartite LCC.

Advertisements

Coloring Warm-Up. A graph is 2-colorable iff it has no odd length cycles 1: If G has an odd-length cycle then G is not 2- colorable Proof: Let v 0, …,

Connectivity - Menger’s Theorem Graphs & Algorithms Lecture 3.

Graphs: basic definitions and notation Definition A (undirected, unweighted) graph is a pair G = (V, E), where V = {v 1, v 2,..., v n } is a set of vertices,

Chapter 8 Topics in Graph Theory

Friendly Partitions of Graphs Jenna Huston Advised by Dr. David Offner.

The Structure of Polyhedra Gabriel Indik March 2006 CAS 746 – Advanced Topics in Combinatorial Optimization.

Course Presentation: A Polynomial Kernel for Multicut in Trees, STACS 2009 Authors: Nicolas Bousquet, Jean Daligault, Stephan Thomasse, Anders Yeo Speaker:

1 Representing Graphs. 2 Adjacency Matrix Suppose we have a graph G with n nodes. The adjacency matrix is the n x n matrix A=[a ij ] with: a ij = 1 if.

k-Factor Factor: a spanning subgraph of graph G

Sub Graph Listing محمد مهدی طالبی دانشگاه صنعتی امیرکبیر.

Mycielski’s Construction Mycielski’s Construction: From a simple graph G, Mycielski’s Construction produces a simple graph G’ containing G. Beginning with.

What is the first line of the proof? a). Assume G has an Eulerian circuit. b). Assume every vertex has even degree. c). Let v be any vertex in G. d). Let.

Randomized Algorithms for the Loop Cutset Problem Author: Ann Becker, Beuven Bar-Yehuda Dan Geiger Beuven Bar-Yehuda Dan Geiger Class presentation for.

NP-Complete Problems Reading Material: Chapter 10 Sections 1, 2, 3, and 4 only.

Matchings Matching: A matching in a graph G is a set of non-loop edges with no shared endpoints Maximal Matching: A maximal matching in a graph is a matching.

Computability and Complexity 24-1 Computability and Complexity Andrei Bulatov Approximation.

Factor Factor: a spanning subgraph of graph G

Factor Factor: a spanning subgraph of graph G k-Factor: a spanning k-regular subgraph Odd component: a component of odd order o(H): the number of odd components.

3/24/03Tucker, Section 4.31 Tucker, Applied Combinatorics, Sec. 4.3, prepared by Jo E-M Bipartite GraphMatching Some Definitions X-Matching Maximal Matching.

What is the next line of the proof? a). Assume the theorem holds for all graphs with k edges. b). Let G be a graph with k edges. c). Assume the theorem.

What is the first line of the backwards direction of this proof? 1.Assume G is a block. 2.Assume every pair of vertices lie on a common cycle. 3.Assume.

Function A function is a relation in which, for each distinct value of the first component of the ordered pair, there is exactly one value of the second.

Homework solution Problem 2. The number of odd degree vertices in a graph is even. (recom. book: G. Harary: Graph Theory) Solution: Let G=(V,E,w) S= 

1/22/03Tucker, Applied Combinatorics, Section EDGE COUNTING TUCKER, APPLIED COMBINATORICS, SECTION 1.3, GROUP B Michael Duquette & Amanda Dargie.

9.2 Graph Terminology and Special Types Graphs

An Illustration of Prim's Algorithm

0.1 Functions and Their Graphs. Real Numbers A set is a collection of objects. The real numbers represent the set of numbers that can be represented as.

On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill.

The Turán number of sparse spanning graphs Raphael Yuster joint work with Noga Alon Banff 2012.

Generalized Derangement Graphs Hannah Jackson.  If P is a set, the bijection f: P  P is a permutation of P.  Permutations can be written in cycle notation.

7.1 and 7.2: Spanning Trees. A network is a graph that is connected –The network must be a sub-graph of the original graph (its edges must come from the.

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

1 Rainbow Decompositions Raphael Yuster University of Haifa Proc. Amer. Math. Soc. (2008), to appear.

Connectivity and Paths 報告人：林清池. Connectivity A separating set of a graph G is a set such that G-S has more than one component. The connectivity of G,

Graph Theory and Applications

Ch 9: Quadratic Equations F) Graphing Quadratic Inequalities Objective: To graph quadratic inequalities.

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

Strings Basic data type in computational biology A string is an ordered succession of characters or symbols from a finite set called an alphabet Sequence.

Optimization Problems Example 1: A rancher has 300 yards of fencing material and wants to use it to enclose a rectangular region. Suppose the above region.

Lecture 52 Section 11.2 Wed, Apr 26, 2006

Basic properties Continuation

Introduction to Graph Theory

Introduction to Graph Theory Lecture 13: Graph Coloring: Edge Coloring.

Introduction to Graph Theory

Graphs Definition: a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected.

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

Angles Project Endiya, Nick, and Mason 5th period Let’s get learning…

English for Economic Informatics I Tomáš Foltýnek Theoretical Foundations of Informatics.

Theory of Computational Complexity Probability and Computing Chapter Hikaru Inada Iwama and Ito lab M1.

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

Proof technique (pigeonhole principle)

Redraw these graphs so that none of the line intersect except at the vertices B C D E F G H.

Graph Theory and Algorithm 01

The Taxi Scheduling Problem

Order Pairs An introduction.

Walks, Paths, and Circuits

Problem Solving 4.

Decision Maths Graphs.

Local Clustering Coefficient

Chapter 1: Linear Functions, Equations, and Inequalities

Euler circuit Theorem 1 If a graph G has an Eulerian path, then it must have exactly two odd vertices. Theorem 2 If a graph G has an Eulerian circuit,

Graph Theory: Cuts and Connectivity

GRAPH THEORY Properties of Planar Graphs Ch9-1.

Dominating Set By Eric Wengert.

Concepts of Computation

GRAPH THEORY Properties of Planar Graphs Ch9-1.

Presentation transcript:

The Grammar According to West By D.B. West

5. Expressions as units. There exists i < j with x i = x j. ( Double-Duty Definition of i, not OK ) There exists i such that i < j and x i = x j. The number of nonneighbors is n-1-d(u)  i. The number of nonneighbors is n-1-d(u), which is at least i. The number of nonneighbors is n-1-d(u), which is greater than or equal to i.

5. Expressions as units. Choose x  V(G) such that x has minimum degree, in (is not a verb, OK) Let G' = G - x. (OK) Let G' = G - x be so-and-so. (not OK) equal (is a verb) Exceptions

5. Expressions as units. Include each vertex independently with probability p=(ln n)/n. ( not OK ) Include each vertex independently with probability p, where p=(ln n)/n. ( OK )

6. Separation of formulas. For x 0. ( not OK ) For x 0. ( OK ) When k=2, G is Eulerian. ( not OK ) When k=2, the graph G is Eulerian. ( OK )

6. Separation of formulas. For every bipartite graph G, χ(G)  2. ( not OK ) If G is bipartite, then χ(G)  2. ( OK )

7. Initial notation. G is so-and-so. ( not OK ) The graph G is so-and-so. ( OK ) Let G be so-and-so. ( OK ) Suppose G is so-and-so. ( OK )

8. Lists of size 2. Let x,y be vertices in G. ( not OK ) My friends John, Mary came to dinner. ( not OK ) Let x and y be vertices in G. (OK ) My friends John and Mary came to dinner. ( OK )

8. Lists of size 2. Since a|b and a,b are maximal and minimal, ( not OK ) Since a|b, with a maximal and b minimal, ( OK ) If x, y are adjacent, ( not OK ) If x and y are adjacent, ( OK ) If {x,y} is a pair of adjacent vertices, ( OK )

8. Lists of size 2. Exceptions. Let x,y,z be the vertices of T, ( OK ) Let {x,y,z} be the vertex set of T, ( OK, more precise ) Let {x,y,z} be a vertex set of T, go to school, go to a school, go to the school, go to church, go to a church, go to the church

8. Lists of size 2. Exceptions. Choose x,y  V(G). ( OK ) For n,m  2, ( not OK ) For n  2 and m  2, ( OK ) For n and m  2, ( not OK ) Suppose n and m are greater than or equal to 2, ( OK )

9. Parenthetic or wordless restrictions. Let m(m  n) be the size. ( not OK ) Let m be the size where m  n. ( OK ) Suppose there is an edge xy (≠e) in G such that… ( not OK ) Suppose there is an edge xy≠e in G such that… ( not OK, Double-Duty Definition ) Suppose that G has an edge xy other than e such that … ( OK )

9. Parenthetic or wordless restrictions. For k  m (k even), ( not OK ) For k  m, k even, ( not OK ) For k  m with k even, ( OK ) Consider a i (1  i  n), ( not OK ) Consider a i for 1  i  n, ( OK )