CMSC 203 / 0201 Fall 2002

1
CMSC 203 / 0201Fall 2002

• Week 13 18/20/22 November 2002
• Prof. Marie desJardins

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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?

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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 (dont 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?

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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?