Graph - Maple Help

PolyhedralSets

 Graph
 generate the graph of the faces for a polyhedral set

 Calling Sequence Graph(polyset) Graph(polyset, output = opts)

Parameters

 polyset - polyhedral set opts - (optional) outputs for function; either plot (default), graph, or a list thereof

Description

 • A directed graph of a polyhedral set can be generated from its faces, called its face lattice.  The nodes in the graph represent the polyhedral set and all of its faces of all possible dimensions, including the empty set which is trivially a face of polyset.  The edges in the graph represent which face is a subset of which, such as vertex that is a subset of the edge that it bounds.
 • The graph is organized into rows, where each row contains faces of polyset that have the same dimension.  The top row contains only polyset itself, followed by its facets below, etc.  The nodes are labelled with an identification number that can be used to retrieve that face using PolyhedralSets[Faces] through the faceid option.
 • The empty set is always included in the graph and labeled as -1, with the other nodes labeled using non-negative integers.
 • The option output = opts controls the return value of the command.  The default value, output = plot, returns the plot of the graph.  The graph object itself can also be obtained using output = [plot, graph].  The returned graph can be used by the commands of the GraphTheory package.

Examples

 > $\mathrm{with}\left(\mathrm{PolyhedralSets}\right):$

Graph of the tetrahedron's faces.  The graph has one node in its top row (the tetrahedron), four nodes in its second row (the tetrahedron's faces), six nodes in its third row (the tetrahedron's edges), four nodes in its fourth row (the tetrahedron's vertices), and one node in its last row (the empty set).

 > $t≔\mathrm{ExampleSets}:-\mathrm{Tetrahedron}\left(\right):$$\mathrm{Graph}\left(t\right)$

The graph object can also be obtained using the output option, which can be further analyzed using the GraphTheory package.

 > $c≔\mathrm{ExampleSets}:-\mathrm{Cube}\left(\right):$$g,p≔\mathrm{Graph}\left(c,'\mathrm{output}'=\left['\mathrm{graph}','\mathrm{plot}'\right]\right):$$p$

The ID's of the facets for a given node can be obtained from the graph.

 > $\mathrm{GraphTheory}\left[\mathrm{Departures}\right]\left(g,154\right)$
 $\left[{109}{,}{91}\right]$ (1)

The PolyhedralSets[Faces] command provides the means of access the faces based on their ID's from the graph.

 > $f≔\mathrm{Faces}\left(c,'\mathrm{faceid}'=91\right)$
 ${f}{≔}{{}\begin{array}{lll}{\mathrm{Coordinates}}& {:}& \left[{{x}}_{{1}}{,}{{x}}_{{2}}{,}{{x}}_{{3}}\right]\\ {\mathrm{Relations}}& {:}& \left[{{x}}_{{3}}{=}{-1}{,}{{x}}_{{2}}{=}{-1}{,}{{x}}_{{1}}{=}{-1}\right]\end{array}$ (2)

Compatibility

 • The PolyhedralSets[Graph] command was introduced in Maple 2015.