# Graph (Network - Nodes and edges)

### Table of Contents

## 1 - About

A graph is a set of vertices connected by edges. See Graph - Graph Model (Network Model)

Data representation that naturally captures complex relationships is a graph (or network).

Except of the special graph that a tree is, the data structure of a graph is non-hierarchical.

Points are called nodes, links are called edges. A link can only connect two nodes (only binary relationship ?)

Each edge has two endpoints, the nodes it connects. The endpoints of an edge are neighbors.

See also: (Graph|Network) - Database

## 2 - Articles Related

## 3 - Application

Are mostly graphs:

- Code (ie Language - (Abstract) Syntax Tree (AST)),
- a workflow editor,
- an organisational chart,
- a business process modelling tool (a UML graph)
- an electronic circuit diagrammer,
- network/telecoms visualisation

## 4 - Type

- Graph - Undirected graph
- (Network|Graph) - Directed acyclic graph (DAG)
- (Network|Graph) - Force
- Graph - Acyclic - graphs that do/don't allow self-loops.
- graphs whose nodes/edges are insertion-ordered, sorted, or unordered

## 5 - Structure

Graph data structure explained: Graph - Data Structure (Physical Representation)

## 6 - Example

### 6.1 - Map

### 6.2 - Directed Graph

(Network|Graph) - Directed acyclic graph (DAG)

- flowcharts
- dependency trees.

## 7 - Dominating set

A dominating set in a graph is a set S of nodes such that every node is in S or a neighbor of a node in S.

Neither algorithm is guaranteed to find the smallest solution.

### 7.1 - Grow Algorithm

```
initialize S = 0;
while S is not a dominating set,
add a node to S.
```

### 7.2 - Shrink Algorithm

```
initialize S = all nodes
while there is a node x such that S −{x} is a dominating set,
remove x from S
```

## 8 - Path

### 8.1 - Definition

### 8.2 - Cycle

A x-to-x path is called a cycle

### 8.3 - Spanning

A set S of edges is spanning for a graph G if, for every edge {x, y} of G, there is an x-to-y path consisting of edges of S.

### 8.4 - Forest

A set of edges of G is a forest if the set includes no cycles.

## 9 - Analysis

See Graph - Analysis