CS 5002: Discrete Math ©Northeastern University Fall 2018 39 Example: Binary Search Trees? Binary search tree (BST) - a tree where nodes are organized in a sorted

In this part, we will study the discrete structures that form the basis of formulating many a real-life problem. The two discrete structures that we will cover are graphs and trees. A graph is a set of points, called nodes or vertices, which are interconnected by a set of lines called edges.

For example, the set {2,4,17,23} is the same as the set {17,4,23,2}. To denote membership we use the ∈ symbol, as in 4 ∈ {2,4,17,23}. On the other hand, non-membership is denoted as in 5 6∈ {2,4,17,23}. If we want to specify a long sequence that follows a pattern, we can use the ellipsis notation, meaning “fill in, using the same ...

- What are graphs? - Graph Examples - Graph Representation - Handshake Theorem - Graph Isomorphism - Connectivity - Graph Coloring - Trees

Consider the following abstract binary relation over universe U = {a,b,c,d,e,f}. R = {(a,b),(b,c),(c,d),(d,e)}. A graph allows us to visualize these relationships. Here is an example of a such graph for this relation: We call the elements a,…,f vertices or nodes of the graph.

EECE 320: Discrete Structures & Algorithms Graphs and Trees Some examples with complete solutions 1. Euler’s Theorem states that for any planar graph G= (V;E) with nvertices, eedges and ffaces, n e+f= 2. Prove using the notion of a dual graph. Proof: Let G be the dual of a graph G. Ghas ffaces and its dual has a vertex for each face in ...

CS340-Discrete Structures Section 1.4 Page 8 Graph Traversals A graph traversal starts at some vertex v and visits all vertices without visiting any vertex more than once. (We assume connectedness: all vertices are reachable from v.) Breadth-First Traversal • First visit v. • Then visit all vertices reachable from v with a path length of 1.

Use adjacency lists to describe the simple graph given in Figure 1. Solution: Table 1 lists those vertices adjacent to each of the vertices of the graph. Represent the directed graph shown in Figure 2 by listing all the vertices that are the terminal …

Discrete Structures •We want to solve problems computationally •This requires –modeling the world –devising an algorithm –determining the efficiency and correctness of that algorithm •Discrete structures: how to model the world and think computationally and rigorously

Example 1: What are the chromatic numbers of the graphs G and H shown in Figure 1? Solution: The chromatic number of G is at least three, because the vertices a, b, and c must be assigned different colors. To see if G can be colored with three colors, assign red to …