Graphs

A data structure that consists of a set of models and a set of edges that relate the nodes to each other.

a graph G isG = (V,E)

where V(G) is a finite, nonempty set of vertices
E(G) is a set of edges (written as pairs of vertices)

  • Vertex: A node in a graph
  • Edge (arc): A pair of vertices representing a connection between two nodes in a graph
  • Undirected graph: A graph in which the edges have no direction



  • Directed graph (digraph): A graph in which each edge is directed from one vertex to another (or the same) vertex





  • Adjacent vertices: Two vertices in a graph that are connected by an edge
  • Path: A sequence of vertices that connects two nodes in a graph

  • loop: path that starts and edges at same vertex

  • simple path - all vertices are distinct with possible exception of first and last being the same.

  • cycle - in a graph where the edges are directed (you can only travel the path along the direction given), is of lenght at least one and the starting and ending vertices are the same.\

  • acyclic - a direct graph that has no cycles (DAG - directed acyclic graph).

  • connected - an undirected graph where there is a path from every vertex to every other vertex.
  • Complete graph: A graph in which every vertex is directly connected to every other vertex



  • Weighted graph: A graph in which each edge carries a value

Graph ADT

Here we present ONLY the basic elements that describe a Graph as part of our ADT.

NOTE: there are no operations to search or traverse a graph....we will not consider this innate to the Graph ADT but, as applications as there are many ways you could implement.

Variables:

  • Vertices - should ideally allow for containing different kinds of information at a vertex.
  • Edges - should allow for direction if needed as well as weight if needed.
 

Operations:

  • Constructor : creates graph instance
  • MakeEmpty: initializes graph to an empty state
  • boolean IsEmpty: tests if graph is empty
  • boolean IsFull: test if graph is full
  • AddVertex( VertexType vertex): Adds vertex to graph if graph is not full.
  • AddEdge( VertexType A, VertexType B, EdgeValueType W): adds edge from A to B with weight W.
  • EdgeValueTypeWeightIs(VertexType A, VertexType B): returns weight of edge from A to B if exists. "null-edge" if no edge exists.
  • GoToVertices(VertexType A, QueType& VertexQueue): returns a queue of vertices adjancent from vertex A.

Other Possible Operations????

These are other possible operations that we will instead leave as a graph applications rather than as innate operations.

  • FindPath(VertexType A, VertexType B) : give a path if one exists from vertex A to vertex B.
  • FindVertex(VertexType A) : returns true or false depending on if a Vertex is found in the graph that matches the content in A.
  • TopologicalSort(Graph g): if g is an acyclic graph (no cylcles) than give the paths from vertices with no incomming edges out to their "ending" vertex.

 

Graph Algorithm Types


Depth-first search algorithm: Visit all the nodes in a branch to its deepest point before moving up  

Breadth-first search algorithm: Visit all the nodes on one level before going to the next level  

Single-source shortest-path algorithm: An algorithm that displays the shortest path from a designated starting node to every other node in the graph

Array-Based Implementation

Adjacency Matrix: for a graph with N nodes, and N by N table that shows the existence (and weights) of all edges in the graph

List-Based Implementation


Adjacency List: A linked list that identifies all the vertices to which a particular vertex is connected; each vertex has its own adjacency list

 

 
© Lynne Grewe