COMP2521

Data Structures & Algorithms

Week 5.1

Graph ADT

Author: Hayden Smith 2021

In this lecture

Why?

  • Graphs are one of the most general and fundamental data types in software, so let's understand it from an abstract point of view

What?

  • Graph Definition
  • Graph Examples
  • Graph Terminology
  • Graph ADT

Graph

A graph G = (V, E) is a general data structure that consists of:

  • V: A set of vertices (i.e. a collection of items)
  • E: A set of edges (relationships between items)

They may contain cycles, and there is no implicit order of items

Graph Examples

Graphs are everywhere:

  • The internet is a graph (vertices = webpage, edge = links)
  • Roads and maps are graphs (vertice = intersections, edge = roads)
  • Trees are graphs (vertices = nodes, edges = children)
    • Trees are a special type of graph with no cycles
  • Linked lists are graphs (vertices = nodes, edges = next links)
    • Linked lists are a special type of tree with 1 child

 

Graph Example (Map)

We can turn a map of Australia into a graph quite easily.

Distance Adelaide Brisbane Canberra Darwin Melbourne Perth Sydney
Adelaide - 2055 1390 3051 732 2716 1605
Brisbane 2055 - 1291 3429 1671 4771 982
Canberra 1390 1291 - 4441 658 4106 309
Darwin 3051 3429 4441 - 3783 4049 4411
Melbourne 732 1671 658 3783 - 3448 873
Perth 2716 4771 4106 4049 3448 - 3972
Sydney 1605 982 309 4411 873 3972 -

Graph Example (Map)

This allows us to ask questions like:

  • What cities are connected to Darwin?
  • Is there a way to get from Brisbane to Perth?

Graph Terminology

  • For neighbours:
    • v1 and v2 are adjacent
    • a is incident on both v1 and v2
  • Degree of a vertex: Number of edges incident
  • Vertex = node (same thing)
  • Edge = link (same thing)

 

Graph Terminology

  • Path: Sequence of vertices where each vertex has an edge to it's predecessor
  • Cycle: A path where the last vertex in the path is the same as the first vertex
  • Length of path: #edges in it

 

Graph Terminology

  • Connected Graph: There is a path from each vertex to every other vertex
    • Most graphs are connected, otherwise they are disjoint
  • Complete Graph: There is an edge from each vertex to every other vertex

 

Graph Terminology

  • Tree: Connected (sub)graph with no cycles
  • Spanning tree: Tree containing all vertices
  • Clique: Complete subgraph

 

Graph Terminology

  • Spanning tree of connected graph G = (V, E):
    • A subgraph of G containing all of V and itself is a single tree (connected, no cycles)
  • Spanning forest of a non-connected graph G = (V, E):
    • A subgraph of G containing all of V, and is a set of trees (not connected, no cycles), with one tree for each connected component

 

Other Graphs

We will look at these, but in later lectures:

  • Directed V Undirected graph
  • Weighted V Unweighted graph

Directed Graph

Weighted Graph

Graph ADT

Core operations:

  • Creating
  • Destroying
  • Inserting vertex/edge
  • Deleting vertex/edge
  • Searching for vertex /edge

 

We will assume simplicity that vertexes are ints.

 

Graph ADT

#include <stdbool.h>

typedef struct GraphRep *Graph;

typedef int Vertex;

typedef struct Edge {
   Vertex v;
   Vertex w;
} Edge;

Graph GraphNew(int);
void  GraphEdgeInsert(Graph, Edge);
void  GraphEdgeRemove(Graph, Edge);
bool  GraphAdjacent(Graph, Vertex, Vertex);
void  GraphShow(Graph);
void  GraphDestroy(Graph);

Graph.h

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