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

September1999 MON 11/18 EQUIVALENCE RELATIONS (6.5)

September1999 October 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

September1999 October 1999 Examples  Exercise 6.5.4: Define three equivalence relations on the set of students in this class.  Exercise : A partition P 1 is a refinement of a partition P 2 if every set in P 1 is a subset of some set in P 2.  (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 R 1 and R 2 are equivalence relations on a set A. Let P 1 and P 2 be the partitions that correspond to R 1 and R 2, respectively. Show that R 1  R 2 iff P 1 is a refinement of P 2.

September1999 October 1999 Examples II  * Exercise : 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 (C 1, C 2 ) is in R iff C 2 can be obtained from C 1 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 11/20 GRAPHS ( )

September1999 October 1999 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

September1999 October 1999 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 K n  Cycle C n (can also say that a graph contains a cycle)  Bipartite graphs, complete bipartite graphs K m, n  Wheels, n-Cubes (don’t need to know these)  Subgraph, union

September1999 October 1999 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?

September1999 October 1999 Examples II  Exercise : The intersection graph of a collection of sets A 1, A 2, …, A n 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) A 1 = {0, 2, 4, 6, 8}, A 2 = {0, 1, 2, 3, 4}, A 3 = {1, 3, 5, 7, 9}, A 4 = {5, 6, 7, 8, 9}, A 5 = {0, 1, 8, 9}  (b) A 1 = {…, -4, -3, -2, -1, 0}, A 2 = {…, -2, -1, 0, 1, 2, …}, A 3 = {…, -6, -4, -2, 0, 2, 4, 6, …}, A 4 = {…, -5, -3, -1, 1, 3, 5, …}, A 5 = {…, -6, -3, 0, 3, 6, …}

September1999 October 1999 Examples III  Exercise : How many vertices and how many edges do the following graphs have?  (a) K n  (b) C n  (d) K m, n  Exercise : How many edges does a graph have if it has vertices of degree 4, 3, 3, 2, 2?  Exercise : How many subgraphs with at least one vertex does K 3 have?

September1999 FRI 11/22 GRAPH STRUCTURE ( )

September1999 October 1999 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

September1999 October 1999 Examples  Exercise 7.3.1/5/26: Represent the given graph with an adjacency list, an adjacency matrix, and an incidence matrix. A CD B

September1999 October 1999 Examples II  Exercise /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 : Show that the following graph is self- complementary. A CD B

September1999 October 1999 Examples III  Exercise (a), (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 : Find all of the cut vertices of the given graphs.  Exercise 7.5.2: Does the graph have an Euler circuit?  Exercise : Can you cross all the bridges exactly once and reurn to the starting point?