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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

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?

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?

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

B

C

D

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

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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