Section 10.1: Graphs and Graph Models. Introduction to Graphs Definition of a Graph Types of Graphs Examples of Graphs

Transcription

1 Graphs Chapter 10

2 Section 10.1: Graphs and Graph Models Introduction to Graphs Definition of a Graph Types of Graphs Examples of Graphs a b c e d

3 Introduction to Graphs Graphs are Discrete Structures that are composed of vertices and edges that connect these vertices This course considers finite graphs- finite number of vertices and edges Types of Graphs - graphs categorized by different properties including: Directed/Undirected Edges Multiple Edges between the same two vertices Loops - Edges that connect a vertex to itself Examples of applications: Flights between cities, or roads within a city Acquaintance Graphs Sports Results Computer Networks and Links between Data Centers

4 Definition of a Graph A graph G = (V, E) consists of V, a nonempty set of vertices (or nodes) and E, a set of edges. Each edge has either one or two vertices associated with it, called its endpoints. An edge is said to connect its endpoints.

5 Definition of a Graph A graph G = (V, E) consists of V, a nonempty set of vertices (or nodes) and E, a set of edges. Each edge has either one or two vertices associated with it, called its endpoints. An edge is said to connect its endpoints. a b

6 Definition of a Graph A graph G = (V, E) consists of V, a nonempty set of vertices (or nodes) and E, a set of edges. Each edge has either one or two vertices associated with it, called its endpoints. An edge is said to connect its endpoints. Example #1 : a 1 b G = (V, E) V = {a, b} E = { 1 }

7 Definition of a Graph Example #2 : G = (V, E) V = {a, b, c} E = { 1, 2 } a 1 b Edges: 1 -> {a, b} 2 -> {b, c} 2 Edge 1 connects vertex a to vertex b c Edge 2 connects vertex b to vertex c Could also write: E = { {a, b}, {b, c} } *note {a, b} = {b, a}

8 Simple Graphs A graph in which each edge connects two different vertices and where no two edges connect the same pair of vertices is called a simple graph.

9 Simple Graphs A graph in which each edge connects two different vertices and where no two edges connect the same pair of vertices is called a simple graph. No Loops: no edge connects a vertex to itself No multiple edges: between two distinct vertices, there exists at most one edge between them Simple Graph a b g Non Simple Graphs: h k d c e i j m n

10 Multigraphs Multiple edges refers to more than one edge connecting the same pair of vertices A graph that may have multiple edges connecting the same vertices are called multigraphs. Example:

11 Pseudographs Loops are edges that connect a vertex to itself Graphs that may contain loops, and possibly multiple edges connecting the same pair of vertices or a vertex to itself, are called pseudographs Examples:

12 Undirected vs Directed Graphs So far we have only considered undirected graphs, as the edges are undirected An edge that connects vertex a with vertex b can be denoted {a, b} or {b, a} as direction is not significant a b Non Simple Graphs: There are also directed graphs where edges can be assigned k a specific direction g h b An edge starting from vertex a to vertex b and will be denoted as an ordered pair (a, b) d c e a i j b m n

13 Directed Graphs A directed graph, or digraph, G = (V, E) consists of a non-empty set of vertices V and a set of directed edges (or arcs) E. Each directed edge is associated with an ordered pair of vertices. The directed edge associated with the ordered pair (u, v) is said to start at u and end at v Examples:

14 Directed Graphs We can obtain a directed graph when we assign a direction to each edge in an undirected graph. Example: d

15 Simple Directed Graph A Simple Directed Graph is a directed graph with no loops and no multiple directed edges Example:

16 Directed Multigraph Directed graphs that may have multiple directed edges from a vertex to a second (possibly the same) vertex Example:

17 Mixed Graph Mixed graphs are graphs with both directed and undirected edges Example:

18 Graph Terminology Type Edges Multiple Edges Allowed? Loops Allowed?