# Graph (Network - Nodes and edges)

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.

## Application

Are mostly graphs:

• a workflow editor,
• an organisational chart,
• a business process modelling tool (a UML graph)
• an electronic circuit diagrammer,
• network/telecoms visualisation

## Type

• Graph - Acyclic - graphs that do/don't allow self-loops.
• graphs whose nodes/edges are insertion-ordered, sorted, or unordered

## Structure

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

## 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.

### Grow Algorithm

``````initialize S = 0;
while S is not a dominating set,
```

### Shrink Algorithm

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

## Path

### Cycle

A x-to-x path is called a cycle

### 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.

### Forest

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