1 Graph Introduction Definitions

2 Definitions I zDirected Graph (or Di-Graph) is an ordered pair G=(V,E) such that yV is a finite, non-empty set (of vertices). yE is a finite set of ordered pairs of vertices. These are the edges of the graph. zUndirected Graph is the same as a directed graph, except that the edges are simply pairs of vertices (no ordering).

3 Definitions II zVertices are the elements of V zEdges are the elements of E zWith respect to an edge e from vertex v to vertex w in a directed graph, v is the tail and w is the head of e; e is said to emanate from v and is incident on w. zThe out-degree v is the number of edges emanating from it; its in-degree is the number of edges incident on it.

4 Definitions III zVertices in an undirected graph simply have a degree. zA Path P is a non-empty sequence of vertices P={v 1, v 2, …, v k } where v i  V for i=1..k, and (v i, v i+1 )  E for i=1..k-1. zThe Path Length is k-1, or the number of edges traversed. zv i+1 is the successor of v i zv i is the predessor of v i+1

5 Definitions IV zA path P is a simple path if no vertex appears more than once in the list of vertices visited. zA Cycle is a path of length > 0 where v 1 =v k zA loop is a cycle of length 1. zA Simple Cycle is a cycle which is also a simple path. zA graph that contains no cycles is acyclic.

6 Definitions V zA DAG is a Directed Acyclic Graph. zA binary tree is a DAG in which the in-degree of each node is one and the out-degree is at most two.

7 The End Slide zText here