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CMSC 203 / 0201 Fall 2002. Week #13 – 18/20/22 November 2002 Prof. Marie desJardins. MON 11/18 EQUIVALENCE RELATIONS (6.5). 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)

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CMSC 203 / 0201 Fall 2002

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Cmsc 203 0201 fall 2002 l.jpg

CMSC 203 / 0201Fall 2002

Week #13 – 18/20/22 November 2002

Prof. Marie desJardins


Mon 11 18 equivalence relations 6 5 l.jpg

MON 11/18 EQUIVALENCE RELATIONS (6.5)


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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


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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.


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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?


Wed 11 20 graphs 7 1 7 2 l.jpg

WED 11/20GRAPHS (7.1-7.2)


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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


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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


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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?


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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, …}


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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?


Fri 11 22 graph structure 7 3 7 5 l.jpg

FRI 11/22GRAPH STRUCTURE (7.3-7.5)


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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


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Examples

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

A

B

C

D


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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

B

C

D


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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?


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