1. 1 Section 8.1 Introduction to Graphs

2. 2 Graph A discrete structure consisting of a set of vertices and a set of edges connecting these vertices –used to model relationships between entities in a variety of fields –used to solve various types of problems

3. 3 Graph types: simple graph Consists of V, a non-empty set of vertices and E, a set of unordered pairs of distinct elements of V Example: V={A,B,C,D,E} E={(A,B),(B,C),(B,E),(E,C),(D,E)} Note that, since pairs are unordered, each could be listed either way A B C D E

4. 4 Graph types: multigraph Consists of a set of vertices (V), a set of edges (E), and a function f from E to {{u,v} | u, v  V, u  v} Edges e 1 and e 2 are called multiple or parallel edges if f(e 1 ) = f(e 2 ) In a simple graph, multiple edges are not allowed, but in a multigraph they are A B C D E

5. 5 Graph types: pseudograph Like a multigraph in which loops are allowed; to formally define, must be able to associate edges to sets containing just one vertex Pseudograph consists of a set V of vertices, a set E of edges, and a function f from E to {{u,v} | u,v  V} An edge is a loop if f(e) = {u,u} = {u} for some u  V

6. 6 Pseudograph Note that multiple edges in a pseudograph may be associated with the same pair of vertices: we say that {u,v} is an edge of graph G = {V,E} if there is at least one edge e with f(e) = {u,v} Example:

7. 7 Undirected graphs Pseudographs are the most general type - can contain loops and multiple edges Multigraphs are undirected graphs that may contain multiple edges but not loops Simple graphs are undirected graphs with no multiple edges or loops

8. 8 Directed Graphs A directed graph (digraph) consists of a set of vertices (V) and a set of edges (E) that are ordered pairs of elements of V Example: V={A,B,C,D} E={(B,A),(B,C),(C,A),(C,D),(D,C)}

9. 9 Directed Multigraph Directed multigraph G=(V,E) consists of a set V or vertices, a set E of edges, and a function f from E to {(u,v)|u,v  V} Edges e 1 and e 2 are multiple edges if f(e 1 )=f(e 2 ) Multiple directed edges are associated to the the same pair of vertices (u,v) is an edge of G=(V,E) as long as there is at least one edge e with f(e) = (u,v)

10. 10 Modeling with graphs - examples Niche overlap graph in ecology: –each species represented by vertex –undirected edge connects 2 vertices if the species compete Influence graph: –each person in group is represented by vertex –directed edge from vertex a to vertex b when a has influence on b

11. 11 Modeling with graphs - examples Round-robin tournament graph: –each team represented by vertex –(a,b) is an edge if team a beats team b Precedence graph: –vertices represent statements in a computer program –directed edge between vertices means 1st statement must be executed before 2nd

12. 12 Section 7.1 Introduction to Graphs