September1999 CMSC 203 / 0201 Fall 2002 Week #13 – 18/20/22 November 2002 Prof. Marie desJardins

September1999

CMSC 203 / 0201Fall 2002

Week #13 – 18/20/22 November 2002

Prof. Marie desJardins

September1999

MON 11/18 EQUIVALENCE RELATIONS (6.5)

September1999October 1999October 1999

Concepts/Vocabulary

Equivalence relation: Relation that is reflexive, symmetric, and transitive (e.g., people born on the same day, strings that are the same length) Equivalence class: Set of all elements “equivalent to” a

given element x (i.e., [x] = {y: (x,y) R}). Partition: disjoint nonempty subsets of S that have S as

their union The equivalence classes of a set form a partition of the

set


Examples

Exercise 6.5.4: Define three equivalence relations on the set of students in this class.

Exercise 6.5.27-28: A partition P1 is a refinement of a partition P2 if every set in P1 is a subset of some set in P2. (27) Show that the partition formed from the congruence

classes modulo 6 is a refinement of the partition formed from the congruence classes modulo 3.

(28) Suppose that R1 and R2 are equivalence relations on a set A. Let P1 and P2 be the partitions that correspond to R1 and R2, respectively. Show that R1 R2 iff P1 is a refinement of P2.


Examples II

* Exercise 6.5.33: Consider the set of all colorings of the 2x2 chessboard where each of the four squares is colored either red or blue. Define the relation R on this set such that (C1, C2) is in R iff C2 can be obtained from C1 either by rotating the chessboard or by rotating it and then reflecting it. (a) Show that R is an equivalence relation. (b) What are the equivalence classes of R?

September1999

WED 11/20GRAPHS (7.1-7.2)


Concepts / Vocabulary [7.1]

Simple graph G = (V, E) – vertices V, edges E A multigraph can have multiple edges between the same

pair of vertices A pseudograph can also have loops (from a vertex to

itself) In an undirected graph, the edges are unordered pairs In a directed graph, the edges are ordered pairs You should be familiar with all of these types of graphs,

but for problem solving, you will only be using simple directed and undirected graphs


Concepts/Vocabulary [6.2]

Adjacent, neighbors, connected, endpoints, incident

Degree of a vertex (number of edges), in-degree, out-degree; isolated, pendant vertices

Complete graph Kn

Cycle Cn (can also say that a graph contains a cycle)

Bipartite graphs, complete bipartite graphs Km, n

Wheels, n-Cubes (don’t need to know these) Subgraph, union


Examples

Exercise 7.1.2: What kind of graph can be used to model a highway system between major cities where (a) there is an edge between the vertices representing

cities if there is an interstate highway between them? (b) there is an edge between the vertices representing

cities for each interstate highway between them? (c) there is an edge between the vertices representing

cities for each interstate highway between them, and there is a loop at the vertex representing a city if there is an interstate highway that circles this city?


Examples II

Exercise 7.1.11: The intersection graph of a collection of sets A1, A2, …, An has a vertex for each set, and an edge connecting two vertices if the corresponding sets have a nonempty intersection. Construct the intersection graph for these sets: (a) A1 = {0, 2, 4, 6, 8}, A2 = {0, 1, 2, 3, 4}, A3 = {1, 3, 5, 7,

9}, A4 = {5, 6, 7, 8, 9}, A5 = {0, 1, 8, 9} (b) A1 = {…, -4, -3, -2, -1, 0}, A2 = {…, -2, -1, 0, 1, 2, …},

A3 = {…, -6, -4, -2, 0, 2, 4, 6, …}, A4 = {…, -5, -3, -1, 1, 3, 5, …}, A5 = {…, -6, -3, 0, 3, 6, …}


Examples III

Exercise 7.2.19: How many vertices and how many edges do the following graphs have? (a) Kn

(b) Cn

(d) Km, n

Exercise 7.2.20: How many edges does a graph have if it has vertices of degree 4, 3, 3, 2, 2?

Exercise 7.2.23: How many subgraphs with at least one vertex does K3 have?

September1999

FRI 11/22GRAPH STRUCTURE (7.3-7.5)


Concepts/Vocabulary

Adjacency list, adjacency matrix, incidence matrix Isomorphism, invariant properties Paths, path length, circuits/cycles, simple paths/circuits Connected graphs, connected components

Strong connectivity, weak connectivity

Cut vertices, cut edges Euler circuit, Euler path Hamilton path, Hamilton circuit

For this section (7.5), need to know terminology but not proofs


Examples

Exercise 7.3.1/5/26: Represent the given graph with an adjacency list, an adjacency matrix, and an incidence matrix.

A

C D

B


Examples II

Exercise 7.3.34/38/41: Determine whether the given pairs of graphs are isomorphic.

A simple graph G is called self-complementary if G and G are isomorphic. Exercise 7.3.50: Show that the following graph is self-

complementary.A

C D

B


Examples III

Exercise 7.3.57(a), 7.3.58(a): Are the simple graphs with the given adjacency matrices / incidence matrices isomorphic?

Exercise 7.4.1: Is the list of vertices a path in the graph? Which paths are circuits? What are the lengths of those that are paths?

Exercise 7.4.15-17: Find all of the cut vertices of the given graphs.

Exercise 7.5.2: Does the graph have an Euler circuit? Exercise 7.5.16: Can you cross all the bridges exactly once

and reurn to the starting point?

Documents

September1999 CMSC 203 / 0201 Fall 2002 Week #13 – 18/20/22 November 2002 Prof. Marie desJardins