# Documentation

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# `Graph`::`topSort`

Topological sorting of the vertices

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## Syntax

```Graph::topSort(`G`)
```

## Description

`Graph::topSort(G)` computes a topological sorting of the graph `G`, i.e., a numbering T of the vertices, such that Ti < Tj whenever there is an edge [i, j] in the graph. Single vertices are positioned at the beginning.

`Graph::topSort` returns a list containing two tables. The first table holds the ordering of the vertices. The second table shows the predecessors of each vertex. If several vertex ui precede a vertex v, the first vertex in the ordering of ui is the predecessor of v. If no predecessor exist, the value will be infinity.

### Note

If `G` contains any cycle then a topological sorting does not exist and the call of `Graph::topSort` results in an error.

## Examples

### Example 1

A "butterfly" graph that is decomposed in three strongly connected components:

```G1 := Graph([a, b, c, d, e, f], [[a, b], [a, c], [a, d], [c, e], [d, e]], Directed): Graph::topSort(G1)```

The first table shows the ordering of the vertices. The left side holds the order for each vertex, whereas the right side holds the name of the vertex. The second table shows the predecessors of each vertex. If no predecessor exist, the right side holds infinity. Otherwise the right side holds the vertex that is the direct predecessor of the vertex on the left side. To see how the graph looks a graphical plotting helps:

```plot(Graph::plotGridGraph(G1, VertexOrder = [None, b, f, a, c, None, None, None, e, None, d, None], VerticesPerLine=3))```

## Parameters

 `G` A graph

## Return Values

List containing two tables.